🧮 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.

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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.

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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.

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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 $(2x - 5)^2$.

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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 $x = 6$

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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.

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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.