Polynomials, Factoring and Quadratics

🧮 Polynomials, Factoring and Quadratics

Part 1 of 7 — Polynomials on the ASVAB: Terms and Like Terms

Topic Overview

What This Looks Like on the ASVAB

Mathematics Knowledge (MK) is the 5th subtest on the CAT-ASVAB. You get 15 scored questions in 31 minutes, so about 2 minutes per question. Every question has 4 answer choices, and no calculator is allowed. You get scratch paper and pencils, and you will use them.

MK is one of the four subtests in your AFQT (AR + MK + 2·VE), the score the military uses to check whether you are eligible to enlist. So every MK point you pick up helps.

The official description of MK includes "algebraic operations and equations," and one of the official sample topics is factoring a difference of squares. That is exactly this lesson. Expect questions like:

• Simplify an expression by combining like terms
• Multiply two binomials
• Factor a trinomial or a difference of squares
• Solve a quadratic equation such as $x^2 - 5x + 6 = 0$

The good news: these questions follow a small number of patterns. Learn the patterns and most of these items take well under 2 minutes.

Warm-up. 🔥 Three quick questions to see where you are. No pressure: Parts 1 to 5 teach all of this.

The Vocabulary You Need

A polynomial is a sum of terms, where each term is a number times variables raised to whole-number powers.

Take $5x^3 - 2x^2 + x - 8$:

Names by number of terms

Like terms

Like terms have exactly the same variable part: the same letters with the same exponents.

Key Fact: The degree is the highest exponent, not the first one you see. In $4 + x - 3x^5$, the degree is $5$ even though that term is written last.

Spot the pieces. 🎯

Adding and Subtracting Polynomials

Adding

Drop the parentheses and combine like terms.

$(2x^2 + 5x - 3) + (4x^2 - x + 7)$

• $x^2$ terms: $2x^2 + 4x^2 = 6x^2$
• $x$ terms: $5x - x = 4x$
• constants: $-3 + 7 = 4$

Answer: $6x^2 + 4x + 4$.

Subtracting: the minus hits EVERY term

A minus sign in front of parentheses changes the sign of every term inside, not just the first one.

$(5x^2 + 3x - 4) - (2x^2 - x + 6)$

Step 1. Distribute the minus: $5x^2 + 3x - 4 - 2x^2 + x - 6$

Step 2. Combine:

• $5x^2 - 2x^2 = 3x^2$
• $3x + x = 4x$
• $-4 - 6 = -10$

Answer: $3x^2 + 4x - 10$.

Watch out: If you change only the first sign, you get $3x^2 + 2x + 2$. That wrong answer is very likely to be one of the four choices.

Evaluating a polynomial

To evaluate, substitute the number in parentheses, then follow the order of operations (exponents before multiplication).

At $x = -3$: $\;x^2 - 4x + 5 = (-3)^2 - 4(-3) + 5 = 9 + 12 + 5 = 26$.

Key Fact: $(-3)^2 = 9$, but $-3^2 = -9$. Always put a negative number in parentheses before you square it.

Add, subtract, evaluate. 🎯

Subtract step by step. 🔽 Simplify $(6y^2 - 2y + 5) - (4y^2 + 3y - 1)$.

Simplify and evaluate. ✍️

Simplify $4m + 9 - m + 2m - 12$ into the form $am + b$.

1) The coefficient $a$:

2) The constant $b$:

3) A separate expression: the value of $x^2 + 5x - 2$ when $x = -4$:

Key Takeaways — Part 1

• MK: 15 questions, 31 minutes, about 2 minutes each, no calculator, 4 choices, and it counts toward your AFQT
• Terms are separated by $+$ and $-$; the sign belongs to the term after it
• Degree = the highest exponent in the polynomial
• Like terms have the same letters with the same exponents; combine them by adding coefficients, and the exponents stay the same
• When subtracting, the minus sign changes every term in the parentheses
• Put negative numbers in parentheses before squaring: $(-3)^2 = 9$

Up Next: Part 2 — multiplying monomials and polynomials, and the FOIL method.

🧮 Polynomials, Factoring and Quadratics

Part 2 of 7 — Multiplying Polynomials and FOIL

Multiplying is where the ASVAB starts to test whether you really know the rules. Three skills cover almost every question:

1. Multiplying monomials (exponent rules)
2. Distributing a monomial across a polynomial
3. Multiplying two binomials with FOIL, including the special products

Skill 1: Multiplying Monomials

Multiply the numbers. For the same variable, add the exponents.

Worked example. $(4x^3)(-3x^2)$

• Numbers: $4 \cdot (-3) = -12$
• Variables: $x^3 \cdot x^2 = x^{3+2} = x^5$
• Answer: $-12x^5$

Worked example. $(2a^2b)(5ab^3)$

• Numbers: $2 \cdot 5 = 10$
• $a^2 \cdot a = a^3$ (a plain $a$ is $a^1$) and $b \cdot b^3 = b^4$
• Answer: $10a^3b^4$

Key Fact: Multiplying powers of the same base → add exponents. A power raised to a power → multiply exponents. Mixing these two up is the most common exponent mistake on the test.

Monomials. 🎯

Skill 2: Distributing a Monomial

Multiply the outside term by every term inside the parentheses.

Worked example. $3x(2x^2 - 5x + 1)$

• $3x \cdot 2x^2 = 6x^3$
• $3x \cdot (-5x) = -15x^2$
• $3x \cdot 1 = 3x$
• Answer: $6x^3 - 15x^2 + 3x$

Worked example with a negative. $-2y(y^2 + 4y - 3)$

• $-2y \cdot y^2 = -2y^3$
• $-2y \cdot 4y = -8y^2$
• $-2y \cdot (-3) = +6y$
• Answer: $-2y^3 - 8y^2 + 6y$

A negative outside flips the sign of every term. Negative times negative is positive.

Skill 3: Multiplying Two Binomials (FOIL)

FOIL is a checklist so you multiply every term of the first binomial by every term of the second:

Combine the middle terms: $-7x + 4x = -3x$. Answer: $x^2 - 3x - 28$.

Worked example. $(2x - 3)(x + 5)$

• F: $2x \cdot x = 2x^2$
• O: $2x \cdot 5 = 10x$
• I: $-3 \cdot x = -3x$
• L: $-3 \cdot 5 = -15$
• Answer: $2x^2 + 7x - 15$

Shortcut for $(x + a)(x + b)$: the answer is $x^2 + (a + b)x + ab$. For $(x + 4)(x - 7)$: $a + b = -3$ and $ab = -28$. This shortcut is also the key to factoring in Part 3.

Distribute. 🎯

FOIL it. 🔽 Multiply $(x - 6)(x + 2)$.

Special Products: Learn These by Sight

Three products show up again and again. Knowing them saves time and helps with factoring later.

Why $(x + 3)^2$ is NOT $x^2 + 9$: squaring means multiplying the binomial by itself. FOIL $(x + 3)(x + 3)$: $x^2 + 3x + 3x + 9 = x^2 + 6x + 9$. The middle term $6x$ is real, and the test will offer you $x^2 + 9$ as a trap.

Why the middle term disappears in $(a + b)(a - b)$: FOIL $(x + 6)(x - 6)$: $x^2 - 6x + 6x - 36$. The outer and inner terms cancel, leaving $x^2 - 36$.

One more: a binomial times a trinomial. Same idea, more terms. Multiply each term of the binomial by each term of the trinomial.

$(x + 2)(x^2 - x + 3)$

• $x(x^2 - x + 3) = x^3 - x^2 + 3x$
• $2(x^2 - x + 3) = 2x^2 - 2x + 6$
• Add: $x^3 + x^2 + x + 6$

Key Fact: $(a + b)^2$ always has a middle term of $2ab$. If a choice squares each piece and stops, it is wrong.

Special products. 🎯

Multiply and match. ✍️

Write $(2x + 3)(x - 4)$ in the form $ax^2 + bx + c$.

1) $a$:

2) $b$:

3) $c$:

Key Takeaways — Part 2

• Same base multiplied → add exponents; power of a power → multiply exponents
• Distribute to every term; a negative outside flips every sign
• FOIL = First, Outer, Inner, Last, then combine the middle terms
• $(x + a)(x + b) = x^2 + (a + b)x + ab$
• $(a \pm b)^2 = a^2 \pm 2ab + b^2$ (never forget the middle term)
• $(a + b)(a - b) = a^2 - b^2$ (the middle terms cancel)

Up Next: Part 3 — running FOIL in reverse: factoring.

🧮 Polynomials, Factoring and Quadratics

Part 3 of 7 — Factoring Out the GCF and Factoring Trinomials

Factoring is multiplying in reverse. Instead of turning $(x + 3)(x + 4)$ into $x^2 + 7x + 12$, you start with $x^2 + 7x + 12$ and find $(x + 3)(x + 4)$.

On the ASVAB, factoring shows up two ways: "Which is the factored form of ...?" and "Which of these is a factor of ...?" It is also the main tool for solving quadratics in Part 5.

Step 1, Always: Pull Out the GCF

The greatest common factor (GCF) is the largest number and the highest power of each variable that divides every term.

Worked example. Factor $6x^3 + 9x^2$.

• Numbers: the GCF of $6$ and $9$ is $3$
• Variables: the smallest power of $x$ is $x^2$
• GCF: $3x^2$
• Divide each term by $3x^2$: $\frac{6x^3}{3x^2} = 2x$ and $\frac{9x^2}{3x^2} = 3$
• Answer: $3x^2(2x + 3)$

Worked example. Factor $8a^2b - 12ab^2$.

• GCF of $8$ and $12$ is $4$; smallest powers are $a$ and $b$
• GCF: $4ab$
• Answer: $4ab(2a - 3b)$

Check by multiplying back: $4ab \cdot 2a = 8a^2b$ and $4ab \cdot (-3b) = -12ab^2$. ✓

Key Fact: "Factor completely" means nothing more can come out. $2x(3x^2 + 6x)$ is not complete, because $3x$ still divides both terms inside.

GCF practice. 🎯

Factoring $x^2 + bx + c$

From Part 2: $(x + p)(x + q) = x^2 + (p + q)x + pq$.

So to factor $x^2 + bx + c$, find two numbers that:

• multiply to $c$ (the constant), and
• add to $b$ (the middle coefficient)

Worked example. $x^2 + 7x + 12$

Answer: $(x + 3)(x + 4)$.

Let the signs do half the work

Worked example. $x^2 - 5x - 24$

• $c = -24$ is negative, so the signs differ; $b = -5$, so the bigger number is negative
• Pairs for $24$: $(1, 24), (2, 12), (3, 8), (4, 6)$. The pair that differs by $5$ is $3$ and $8$
• Numbers: $-8$ and $+3$ (product $-24$, sum $-5$)
• Answer: $(x - 8)(x + 3)$

GCF first, then the trinomial

$2x^2 - 6x - 20 = 2(x^2 - 3x - 10) = 2(x - 5)(x + 2)$

When the $x^2$ coefficient is not 1

For something like $2x^2 + 7x + 3$, the first terms must be $2x$ and $x$. Try placing the factors of $3$ and check the middle term with FOIL:

• $(2x + 3)(x + 1)$: middle term $2x + 3x = 5x$ ✗
• $(2x + 1)(x + 3)$: middle term $6x + x = 7x$ ✓

Answer: $(2x + 1)(x + 3)$.

Key Fact: You can always check a factoring answer by multiplying it back out. On a multiple-choice test, that check takes about 15 seconds.

Factor the trinomial. 🎯

GCF, then trinomial. 🔽 Factor $3x^2 - 12x - 36$ completely.

Fill in the factors. ✍️

1) $2x^2 + 11x + 5 = (2x + a)(x + b)$. What is $a$?

2) What is $b$?

3) $x^2 - 2x - 35 = (x - p)(x + q)$, with $p$ and $q$ positive. What is $p$?

4) What is $q$?

Key Takeaways — Part 3

• Always take out the GCF first: the biggest number and the smallest power of each variable that every term shares
• $x^2 + bx + c$: find two numbers that multiply to $c$ and add to $b$
• Signs: $c$ positive → same signs (matching $b$); $c$ negative → opposite signs, larger number takes the sign of $b$
• If the $x^2$ coefficient is not $1$, try the factor placements and check the middle term
• Multiply back to check any factoring answer

Up Next: Part 4 — two patterns you can factor on sight, and simplifying algebraic fractions.

🧮 Polynomials, Factoring and Quadratics

Part 4 of 7 — Special Patterns and Algebraic Fractions

Two factoring patterns are worth memorizing because you can spot them in seconds. One of them, the difference of squares, is named in the official MK sample topics.

Pattern 1: Difference of Squares

$a^2 - b^2 = (a + b)(a - b)$

It is the "sum times difference" product from Part 2, run backward. Look for: two terms, both perfect squares, with a minus sign between them.

Hard version. $x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)$. The piece $x^2 - 4$ is another difference of squares, so keep going.

Watch out: A sum of squares, like $x^2 + 25$, does not factor with whole numbers. And $(x - 5)^2$ is not the same as $x^2 - 25$; it equals $x^2 - 10x + 25$.

A mental-math trick

Difference of squares turns ugly arithmetic into easy arithmetic, which matters when you have no calculator:

$51^2 - 49^2 = (51 + 49)(51 - 49) = 100 \cdot 2 = 200$

Pattern 2: Perfect-Square Trinomials

$a^2 + 2ab + b^2 = (a + b)^2 \qquad a^2 - 2ab + b^2 = (a - b)^2$

Test: the first and last terms are perfect squares, and the middle term is 2 times their square roots.

• $x^2 + 12x + 36$: $\sqrt{x^2} = x$, $\sqrt{36} = 6$, and $2 \cdot x \cdot 6 = 12x$ ✓. So it is $(x + 6)^2$.
• $4x^2 - 20x + 25$: roots $2x$ and $5$, and $2 \cdot 2x \cdot 5 = 20x$ ✓. Middle sign is minus, so $(2x - 5)^2$.

Spot the pattern. 🎯

Algebraic Fractions

Dividing monomials: subtract exponents

$\frac{x^7}{x^3} = x^{7-3} = x^4 \qquad \frac{6x^3}{2x} = 3x^2 \qquad \frac{10a^4b^2}{5ab^2} = 2a^3$

Divide the numbers, subtract exponents for each variable. In the last one, $\frac{b^2}{b^2} = 1$, so $b$ disappears.

Simplifying a fraction of polynomials

Factor the top and the bottom, then cancel common FACTORS.

Worked example. $\frac{x^2 - 9}{x + 3}$

• Top: $(x + 3)(x - 3)$
• Cancel the common factor $(x + 3)$
• Answer: $x - 3$ (for $x \neq -3$)

Worked example. $\frac{x^2 + 5x + 6}{x^2 - 4}$

• Top: $(x + 2)(x + 3)$. Bottom: $(x + 2)(x - 2)$
• Cancel $(x + 2)$
• Answer: $\frac{x + 3}{x - 2}$

The cancelling trap

You may cancel only things that are multiplied, never pieces that are added.

Test it with a number: at $x = 2$, $\frac{x + 6}{6} = \frac{8}{6}$, not $2$.

Key Fact: Factor first, cancel second. If you cannot write the top and bottom as products, you cannot cancel anything.

Simplify. 🎯

Factor, then cancel. 🔽 Simplify $\frac{x^2 - 16}{x^2 + 2x - 8}$.

Cancel and evaluate. ✍️

1) $\frac{x^2 + 7x + 10}{x + 5} = x + k$. What is $k$?

2) Simplify $\frac{x^2 - 81}{x - 9}$, then find its value at $x = 11$:

3) Without a calculator: $35^2 - 15^2 =$

Key Takeaways — Part 4

• Difference of squares: $a^2 - b^2 = (a + b)(a - b)$. Two square terms with a minus between them
• A sum of squares ($x^2 + 25$) does not factor
• Perfect-square trinomial: first and last terms are squares and the middle is $2ab$ → $(a \pm b)^2$
• Dividing monomials: divide the numbers, subtract the exponents
• Algebraic fractions: factor first, then cancel common factors, never added terms
• $a^2 - b^2 = (a + b)(a - b)$ is also a no-calculator arithmetic shortcut

Up Next: Part 5 — using factoring to solve quadratic equations.

🧮 Polynomials, Factoring and Quadratics

Part 5 of 7 — Solving Quadratics by Factoring

A quadratic equation has an $x^2$ term and no higher power, like $x^2 - 5x + 6 = 0$. Most quadratics have two solutions, and the ASVAB usually lists both in each answer choice ("$x = 2$ or $x = 3$").

The Zero-Product Property

If two numbers multiply to zero, at least one of them is zero:

$\text{If } A \cdot B = 0, \text{ then } A = 0 \text{ or } B = 0.$

That is why factoring solves quadratics: it turns one hard equation into two easy ones.

The Four Steps

1. Move everything to one side so the other side is $0$
2. Factor
3. Set each factor equal to $0$ and solve
4. Check (plug a solution back in, or check an answer choice)

Worked example. $x^2 - 5x + 6 = 0$

• Factor: $(x - 2)(x - 3) = 0$
• $x - 2 = 0$ gives $x = 2$; $x - 3 = 0$ gives $x = 3$
• Check $x = 2$: $4 - 10 + 6 = 0$ ✓

Worked example: not set to zero yet. $x^2 + 4x = 21$

• Subtract $21$: $x^2 + 4x - 21 = 0$
• Factor: $(x + 7)(x - 3) = 0$
• Solutions: $x = -7$ or $x = 3$

Sign check: the factor $(x + 7)$ gives the solution $x = -7$. The solution always has the opposite sign of the number in the factor.

Worked example: a common factor of $x$. $x^2 = 6x$

• Do not divide both sides by $x$. That throws away a solution
• Move it over: $x^2 - 6x = 0$
• Factor: $x(x - 6) = 0$
• Solutions: $x = 0$ or $x = 6$

Solve by factoring. 🎯

Two More Types

$x^2 = k$: take the square root, and remember $\pm$

$x^2 = 49$ means $x = 7$ or $x = -7$, because $(-7)^2 = 49$ too. Write it as $x = \pm 7$.

Worked example. $3x^2 - 75 = 0$

• Add $75$: $3x^2 = 75$
• Divide by $3$: $x^2 = 25$
• $x = \pm 5$

(Factoring works too: $3(x^2 - 25) = 3(x + 5)(x - 5) = 0$.)

When the $x^2$ coefficient is not 1

Worked example. $2x^2 + 5x - 3 = 0$

• Factor: $(2x - 1)(x + 3) = 0$ (check: $2x^2 + 6x - x - 3$ ✓)
• $2x - 1 = 0$ gives $x = \frac{1}{2}$; $x + 3 = 0$ gives $x = -3$

Word Problems That Lead to a Quadratic

Worked example. A rectangular storage pad is $3$ feet longer than it is wide, and its area is $40$ square feet. Find the width.

• Let the width be $w$. Then the length is $w + 3$
• Area: $w(w + 3) = 40$
• Expand and set to zero: $w^2 + 3w - 40 = 0$
• Factor: $(w + 8)(w - 5) = 0$, so $w = -8$ or $w = 5$
• A width cannot be negative, so $w = 5$ feet (and the length is $8$ feet)

Key Fact: In a word problem, check which solutions make sense. Lengths, ages and counts cannot be negative, so throw those out.

Sum and product of the solutions

If $(x - 2)(x + 7) = 0$, the solutions are $2$ and $-7$. Their sum is $-5$ and their product is $-14$. Questions that ask for the sum or product of the solutions are just solving plus one extra step.

Set up and solve. 🔽 A rectangular tent floor is $4$ feet longer than it is wide, and its area is $60$ square feet.

One step further. 🎯

Find the solutions. ✍️

1) The positive solution of $4x^2 = 100$:

2) $3x^2 + 10x - 8 = 0$ has one positive and one negative solution. The positive solution (a fraction is fine):

3) The negative solution of $3x^2 + 10x - 8 = 0$:

Key Takeaways — Part 5

• Zero-product property: if $A \cdot B = 0$, then $A = 0$ or $B = 0$
• Steps: set to $0$ → factor → each factor $= 0$ → check
• The factor $(x + 7)$ gives $x = -7$: the solution has the opposite sign
• Never divide both sides by $x$; factor it out instead, and keep $x = 0$
• $x^2 = k$ gives $x = \pm\sqrt{k}$ (two answers)
• In word problems, throw out solutions that make no sense (negative lengths)

Up Next: Part 6 — strategy for this topic on test day, and the traps the answer choices are built from.

🧮 Polynomials, Factoring and Quadratics

Part 6 of 7 — Test Strategy and Common Traps

Your Time Budget

MK gives you 31 minutes for 15 questions, about 2 minutes each. A polynomial question you recognize (FOIL, a difference of squares, a simple trinomial) should take 30 to 60 seconds. That banks time for the harder items.

Three things about the CAT-ASVAB change how you work:

• You cannot go back. Once you answer, the question is gone. Do your quick check before you click.
• It adapts. A right answer usually brings a harder question. Seeing harder questions is a good sign, not a reason to panic.
• Don't rush-guess at the end. If time runs short, keep genuinely working each question. The CAT applies a relatively large penalty when several wrong answers come at the end of a subtest, and unfinished items are scored as if you guessed at random.

Some questions may be unscored tryout items, but you can't tell which, so treat every question as if it counts.

Four Strategies That Work Without a Calculator

1. Multiply back. For "Which is the factored form?", FOIL each choice until one matches. Usually the first or second choice you test settles it.

2. Plug the choices into the equation. For "Solve $x^2 - x - 12 = 0$", try a choice: $x = 4$ gives $16 - 4 - 12 = 0$ ✓. Remember that both values in a choice must work.

3. Plug in a number to test "equivalent" questions. Pick a value like $x = 2$ or $x = 3$, evaluate the original and each choice, and keep the choice that matches. Avoid $x = 0$ and $x = 1$: too many wrong choices match by accident. Also avoid a value that makes the expression $0$.

4. Take out the GCF first. It makes every number smaller and shows up in the answer choices.

The Traps the Wrong Answers Are Built From

ASVAB distractors are not random. Each one is the result of a specific, common mistake. Know the mistakes and you can spot the traps.

Name the trap. 🪤

Diagnose the error. 🔽 Each line of student work has one mistake. Name it.

Check before you click. 🎯

Plug-in test. ✍️

A question asks which expression equals $2x^2 - 5x - 3$. Test with $x = 2$.

1) The value of $2x^2 - 5x - 3$ at $x = 2$:

2) The value of the choice $(2x - 1)(x + 3)$ at $x = 2$:

3) The value of the choice $(2x + 1)(x - 3)$ at $x = 2$:

Key Takeaways — Part 6

• Budget about 2 minutes per MK question; known polynomial patterns should take under a minute
• The CAT does not let you go back: check before you click
• Strategies: multiply back, plug in the choices, plug in a number (not $0$ or $1$), GCF first
• Every wrong choice comes from a known mistake: missing middle term, one-term minus, exponent rule mix-ups, cancelling terms, dividing by $x$, sign flips, missing $\pm$
• If time runs short, keep working each question honestly rather than guessing in a string

Up Next: Part 7 — a mixed mastery set across the whole topic.

🧮 Polynomials, Factoring and Quadratics

Part 7 of 7 — Mixed Mastery Set

Time to put it all together. These 12 questions mix every skill from Parts 1 to 5, in roughly the mix you will see on test day: a few quick ones, a solid middle, and a few that take real work.

Test conditions: no calculator, scratch paper only, and aim for about 2 minutes per question. Do your check before you answer, just as you would on the CAT-ASVAB, where you can't go back.

Set A: warm-up speed. ⏱️

Set B: the core of the test. 🎯

Set C: the harder questions. 🏋️

Final challenge. ✍️

Multiply $(x + 6)(x^2 - 2x + 1)$ and write the result as $x^3 + ax^2 + bx + c$.

1) $a$:

2) $b$:

3) $c$:

Key Takeaways — Polynomials, Factoring and Quadratics

• Like terms combine by their coefficients; subtracting flips every sign
• Multiply powers → add exponents; power of a power → multiply exponents
• FOIL every pair of binomials; $(a \pm b)^2$ has a middle term of $\pm 2ab$
• Factor: GCF first, then two numbers that multiply to $c$ and add to $b$
• Difference of squares $a^2 - b^2 = (a + b)(a - b)$; a sum of squares does not factor
• Algebraic fractions: factor, then cancel factors only
• Solve quadratics: set to $0$, factor, set each factor to $0$, flip the sign, keep $x = 0$ and $\pm$ answers, check

You now have the whole toolkit for polynomial questions on MK. Review any part where you missed a question, then try the exit quiz.