Multiplying Polynomials

✖️ Multiplying Polynomials

Part 1 of 5 — The Distributive Property & Exponents

Topics in This Part

🔑 Key Concept: Every polynomial multiplication — no matter how big — is just the distributive property applied carefully: each term in the first factor multiplies each term in the second.

A Quick Vocabulary Refresher

A polynomial is a sum of terms, where each term is a number times a power of $x$ (with whole-number exponents).

The number in front of a term is its coefficient; the exponent on $x$ tells you the degree of that term.

💡 In $-7x^3$, the coefficient is $-7$ and the degree is $3$. The sign belongs to the coefficient — keep it attached as you multiply.

Multiplying Monomials

To multiply two single terms, do two separate jobs:

1. Multiply the coefficients (regular number multiplication).
2. Add the exponents on matching variables: $x^m \cdot x^n = x^{m+n}$.

Worked Example

$(3x^2)(5x^4) = (3 \cdot 5)\,(x^{2} \cdot x^{4}) = 15x^{6}$

The coefficients $3$ and $5$ multiply to $15$; the exponents $2$ and $4$ add to $6$.

Another

$(-4x)(2x^3) = (-4 \cdot 2)\,(x^{1} \cdot x^{3}) = -8x^{4}$

⚠️ Add exponents when multiplying — don't multiply them. $x^2 \cdot x^4 = x^6$, not $x^8$.

Concept Check 🎯

Multiply the Monomials 🧮

Enter each simplified product. Write powers with a caret, e.g. 12x^5.

1) $(4x^2)(7x^3) = \,?$ 2) $(6x)(x^5) = \,?$ 3) $(-3x^4)(-2x^2) = \,?$

Distributing a Monomial Over a Sum

The distributive property says $a(b + c) = ab + ac$. With polynomials it works exactly the same — just apply the monomial rule to each term.

Worked Example

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

Distribute $3x$ to all three terms:

$= 3x \cdot x^2 + 3x \cdot 4x + 3x \cdot (-5)$ $= 3x^3 + 12x^2 - 15x$

🔑 Pattern to remember: A monomial times an $n$-term polynomial gives an $n$-term answer. One term in, three terms out — nothing should disappear.

We'll lean on this exact move for the rest of the lesson.

Distribute the Monomial 🔽

Expand $2x(3x^2 + 5x - 1)$ one term at a time.

✖️ Multiplying Polynomials

Part 2 of 5 — Monomial × Polynomial, Carefully

🔑 The Idea: Multiplying a monomial across a polynomial is reliable if you (1) hit every term, (2) keep every sign, and (3) combine exponents correctly. Part 2 makes that automatic.

Don't Lose the Signs

The most common error is mishandling negatives. Treat the sign as part of each term before you distribute.

Worked Example: $-2x(3x^2 - 4x + 7)$

Distribute $-2x$ to each term, signs included:

$-2x \cdot 3x^2 = -6x^3$ $-2x \cdot (-4x) = +8x^2$ $-2x \cdot 7 = -14x$

Combine:

$-2x(3x^2 - 4x + 7) = -6x^3 + 8x^2 - 14x$

⚠️ Notice the middle term flipped to positive: a negative monomial times a negative term gives a positive product. Track signs term by term.

Distribute Term by Term 🔽

You're expanding $4x(2x^2 - 3x + 5)$. Choose each product.

More Than One Variable

When terms contain two variables, add exponents separately for each letter.

Worked Example: $2xy(3x^2 - 5y)$

$2xy \cdot 3x^2 = 6x^{3}y \qquad (x^1 \cdot x^2 = x^3,\ y \text{ carries along})$ $2xy \cdot (-5y) = -10xy^{2} \qquad (y^1 \cdot y^1 = y^2,\ x \text{ carries along})$

$2xy(3x^2 - 5y) = 6x^3y - 10xy^2$

💡 A variable that appears in only one factor just "rides along" into the product with the exponent it already had.

Concept Check 🎯

Expand and Simplify 🧮

Enter the coefficient asked for in each fully-expanded product.

1) In $2x(4x^2 - x + 6)$, the coefficient of $x^3$ is $\,?$ 2) In $-x(x^2 - 9x + 2)$, the coefficient of $x^2$ is $\,?$ 3) In $3x(2x - 7)$, the constant times $x$ (the coefficient of $x$) is $\,?$

✖️ Multiplying Polynomials

Part 3 of 5 — Binomial × Binomial (FOIL)

🔑 The Big One: Multiplying two binomials is the skill you'll use most. FOIL is just a way to make sure you hit all four products: each term of the first binomial times each term of the second.

What FOIL Stands For

For $(a + b)(c + d)$, FOIL names the four products:

$(a+b)(c+d) = ac + ad + bc + bd$

💡 FOIL is just the distributive property done twice. It only has a name because two binomials come up so often. For bigger polynomials we'll drop the acronym and distribute directly (Part 5).

Worked Example: $(x + 3)(x + 5)$

$\textbf{F: } x \cdot x = x^2$ $\textbf{O: } x \cdot 5 = 5x$ $\textbf{I: } 3 \cdot x = 3x$ $\textbf{L: } 3 \cdot 5 = 15$

Add them, then combine the like middle terms $5x + 3x$:

$(x+3)(x+5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15$

🔑 Almost always, the Outer and Inner products are like terms and combine into the middle term. A clean binomial-times-binomial gives a trinomial.

Concept Check 🎯

Watch the Signs in FOIL

Subtraction inside a binomial just means a negative term. Treat $(x - 4)$ as $(x + (-4))$.

Worked Example: $(2x - 3)(x - 5)$

$\textbf{F: } 2x \cdot x = 2x^2$ $\textbf{O: } 2x \cdot (-5) = -10x$ $\textbf{I: } -3 \cdot x = -3x$ $\textbf{L: } -3 \cdot (-5) = +15$

$(2x-3)(x-5) = 2x^2 - 10x - 3x + 15 = 2x^2 - 13x + 15$

⚠️ The Last product is $(-3)(-5) = +15$, a positive constant. Two negatives in the last terms always give a positive constant.

Build the FOIL 🔽

Expand $(3x + 2)(x - 4)$ one product at a time.

FOIL Practice 🧮

Each product is a trinomial $ax^2 + bx + c$. Enter the middle coefficient $b$ after combining like terms.

1) $(x + 4)(x + 9)$: $\;b = \,?$ 2) $(x - 6)(x + 2)$: $\;b = \,?$ 3) $(2x + 1)(3x - 5)$: $\;b = \,?$

✖️ Multiplying Polynomials

Part 4 of 5 — Special Products (Shortcuts)

🔑 Why Bother: Two binomial products show up constantly — the square of a binomial and the difference of squares. Learning their patterns lets you skip FOIL and write the answer directly.

Squaring a Binomial

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

The pattern: square the first, square the last, and the middle is twice the product of the two terms.

Worked Example: $(x + 5)^2$

$a = x,\ b = 5 \;\Rightarrow\; x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25$

Worked Example: $(x - 4)^2$

$x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16$

⚠️ The #1 mistake: $(x+5)^2 \ne x^2 + 25$. You must include the middle term $2ab$. Squaring a sum is not the same as squaring each piece.

Concept Check 🎯

Difference of Squares

When the two binomials are the same except for the sign, the middle terms cancel:

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

Worked Example: $(x + 6)(x - 6)$

$\textbf{O+I: } -6x + 6x = 0 \quad\text{(they cancel!)}$ $(x+6)(x-6) = x^2 - 36$

Worked Example: $(3x - 2)(3x + 2)$

$(3x)^2 - (2)^2 = 9x^2 - 4$

💡 How to spot it: same first term, same last term, but one $+$ and one $-$. The answer is always "first squared minus last squared" — no middle term at all.

Pick the Pattern 🔽

Identify each product without doing full FOIL.

Use the Shortcut 🧮

Enter the requested value using the special-product patterns.

1) In $(x + 10)^2 = x^2 + bx + c$, the middle coefficient $b = \,?$ 2) $(x - 12)(x + 12) = x^2 - c$. Enter $c$. $\,?$ 3) In $(4x + 3)^2 = 16x^2 + bx + 9$, the middle coefficient $b = \,?$

✖️ Multiplying Polynomials

Part 5 of 5 — Bigger Products, Applications & Mastery Check

You can now multiply monomials, distribute across any polynomial, FOIL two binomials, and use special-product shortcuts. The last skill: multiplying larger polynomials and applying it to area.

Binomial × Trinomial

There's no new rule — just distribute every term of the first across every term of the second, then combine like terms. A binomial times a trinomial means $2 \times 3 = 6$ products before combining.

Worked Example: $(x + 2)(x^2 + 3x + 4)$

Distribute the $x$, then the $2$:

$x(x^2 + 3x + 4) = x^3 + 3x^2 + 4x$ $2(x^2 + 3x + 4) = 2x^2 + 6x + 8$

Now stack and combine like terms:

$x^3 + (3x^2 + 2x^2) + (4x + 6x) + 8 = x^3 + 5x^2 + 10x + 8$

🔑 Stay organized: line up like powers in columns. The degree of the answer is the sum of the degrees of the factors — here $1 + 2 = 3$, so a cubic.

Concept Check 🎯

Application: Area of a Rectangle

If a rectangle has width $(x + 3)$ and length $(x + 7)$, its area is the product:

$A = (x + 3)(x + 7) = x^2 + 10x + 21$

This is exactly a FOIL — the polynomial is the area written in terms of $x$.

A "Bordered" Example

A square photo has side $x$. You add a $2$-inch mat on every side, so the framed square has side $(x + 4)$ (two inches on each of two sides). Its area:

$(x + 4)^2 = x^2 + 8x + 16$

💡 Polynomial multiplication is how you turn a picture's dimensions into a single formula you can evaluate for any $x$.

Area Practice 🧮

A rectangle has width $(x + 5)$ and length $(2x + 1)$. Its area is $2x^2 + bx + c$.

1) Find the middle coefficient $b$. $\,?$ 2) Find the constant $c$. $\,?$ 3) Evaluate the area when $x = 3$ (a plain number). $\,?$

Exit Quiz ✅

Answer all three to finish the lesson.