Expanding and Combining Polynomials, SAT math lesson

Picture a rectangular garden that is $x + 3$ meters long and $x + 5$ meters wide. Cut it into four smaller rectangles along the $x$ marks and the areas are $x^2$ , $5x$ , $3x$ and $15$ . Add them and you get $x^2 + 8x + 15$ , which is the same area written as a sum instead of a product. Expanding is that bookkeeping done in algebra, and many SAT questions about equivalent expressions are really tests of whether every piece got counted.

See also Expanding and Factoring for a Purpose in Foundations for Calculus.

Key terms

Polynomial

Definition A polynomial in $x$ is an expression that can be written as a finite sum of terms of the form $a x^k$ , where each coefficient $a$ is a real number and each exponent $k$ is a nonnegative integer,

a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0.

Expressions with $x$ in a denominator, under a radical or in an exponent, such as $\frac{1}{x}$ , $\sqrt{x}$ or $2^x$ , are not polynomials in $x$ . The same idea applies with several variables, where each term is a real number times a product of nonnegative integer powers of the variables.

In plain words A polynomial is built only from numbers and the variable by adding, subtracting and multiplying. No dividing by the variable and no roots of it.

Example The expression $4x^3 - 2x + 7$ is a polynomial with coefficients $4$ , $0$ , $-2$ and $7$ for $x^3$ , $x^2$ , $x$ and the constant. The expression $x^2 + \frac{3}{x}$ is not a polynomial, because $\frac{3}{x} = 3x^{-1}$ has a negative exponent.

Degree

Definition The degree of a nonzero polynomial in $x$ is the largest exponent $k$ whose term $a_k x^k$ has a nonzero coefficient, after like terms are combined. A nonzero constant has degree $0$ , and the zero polynomial is usually given no degree. For nonzero polynomials $p$ and $q$ ,

\deg(pq) = \deg p + \deg q.

In plain words Degree is the highest power that actually survives. Combine terms first, because a high power can cancel.

Example The product $(2x^3 + 1)(x^2 - 4)$ has degree $3 + 2 = 5$ , and its leading term is $2x^3 \cdot x^2 = 2x^5$ . By contrast, $(x^2 + 3x) - (x^2 - 1)$ looks like degree $2$ , but it simplifies to $3x + 1$ , which has degree $1$ .

Like terms

Definition Two terms are like terms when they contain the same variables raised to the same powers, so they differ at most in their numerical coefficients. Like terms combine by adding coefficients,

a x^k + b x^k = (a + b) x^k.

Terms with different powers, such as $3x^2$ and $3x$ , are not like terms and cannot be combined into one term.

In plain words You can add apples to apples. Here $x^2$ terms go with $x^2$ terms and $xy$ terms go with $xy$ terms, and nothing else merges.

Example Simplify $(5x^2 - 3x + 4) - (2x^2 + x - 6)$ . The minus sign applies to every term in the second group,

\begin{aligned} &5x^2 - 3x + 4 - 2x^2 - x + 6 \\[0.4em] &= 3x^2 - 4x + 10. \end{aligned}

A common slip is to subtract only the $2x^2$ and keep $+x - 6$ , which gives the wrong result $3x^2 - 2x - 2$ .

Distributive property

Definition For all real numbers $a$ , $b$ and $c$ ,

a(b + c) = ab + ac.

Applied repeatedly, it multiplies a single term by every term of a sum, $a(b_1 + b_2 + \cdots + b_k) = ab_1 + ab_2 + \cdots + ab_k$ , and a negative factor changes the sign of every term it multiplies.

In plain words The number outside the parentheses has to visit every term inside. When it is negative, every sign inside flips.

Example Expanding $-3x(2x^2 - x + 5)$ term by term gives

-6x^3 + 3x^2 - 15x.

The middle term is positive because $(-3x)(-x) = 3x^2$ .

Product of polynomials

Definition To multiply two polynomials, multiply every term of the first by every term of the second and add all the results. If the factors have $m$ and $n$ terms, there are $m \cdot n$ products before like terms are combined. For two binomials,

\begin{aligned} (a + b)(c + d) &= ac + ad \\[0.4em] &\quad + bc + bd. \end{aligned}

When $a$ , $b$ , $c$ and $d$ are positive lengths, this is the area of a rectangle with sides $a + b$ and $c + d$ split into four smaller rectangles. That picture only makes sense for nonnegative lengths, but the identity itself holds for all real numbers, because it follows from the distributive property alone.

In plain words Every term meets every other term exactly once. With signed terms, keep each sign attached to its term rather than imagining negative lengths.

Example The garden from the opening, with $x > 0$ , has area

\begin{aligned} &(x + 3)(x + 5) \\[0.4em] &= x^2 + 5x + 3x + 15 \\[0.4em] &= x^2 + 8x + 15. \end{aligned}

With a negative term the same four products appear,

\begin{aligned} &(2x - 3)(x + 4) \\[0.4em] &= 2x^2 + 8x - 3x - 12 \\[0.4em] &= 2x^2 + 5x - 12. \end{aligned}

Squaring a binomial also needs all four products. The result is $(x + 4)^2 = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$ , and the shortcut $x^2 + 16$ drops the two middle products.

Equivalent expressions

Definition Two expressions in $x$ are equivalent when they have the same value for every value of $x$ for which both are defined. For polynomials that means every real $x$ . Expanding, combining like terms and factoring produce equivalent expressions.

One value of $x$ where two expressions disagree proves they are not equivalent. One value where they agree proves nothing, since different expressions can match at a few points.

In plain words Equivalent means equal everywhere, not just at one lucky number. Plugging in a number is a fast way to rule out wrong answer choices.

Example Is $(x + 1)^2$ equivalent to $x^2 + 1$ ? At $x = 0$ both equal $1$ , which settles nothing. At $x = 1$ the first is $4$ and the second is $2$ , so they are not equivalent. The correct expansion is $x^2 + 2x + 1$ .

Coefficient matching

Definition Two polynomials in $x$ are equal for every real number $x$ exactly when, after each is fully expanded and like terms are combined, they have the same coefficient for each power of $x$ . So if

\begin{gathered} a_n x^n + \cdots + a_1 x + a_0 \\[0.4em] = b_n x^n + \cdots + b_1 x + b_0 \end{gathered}

for all real $x$ , then $a_k = b_k$ for every $k$ .

In plain words When an SAT question says two expressions are equal for all values of $x$ , expand both sides and set the $x^2$ numbers equal, the $x$ numbers equal and the constants equal.

Example Suppose $(ax + 3)(2x - 1) = 6x^2 + bx - 3$ for all $x$ , where $a$ and $b$ are constants. Expanding the left side gives

2a x^2 + (6 - a) x - 3.

Matching the $x^2$ coefficients gives $2a = 6$ , so $a = 3$ . Matching the $x$ coefficients gives $b = 6 - a = 3$ . The constants already agree, since both are $-3$ . A check confirms it, because $(3x + 3)(2x - 1) = 6x^2 - 3x + 6x - 3 = 6x^2 + 3x - 3$ .

Check yourself

Which expression is equivalent to $(3x - 2)(x + 5) - x(x - 4)$ ? (A) $2x^2 + 17x - 10$ (B) $2x^2 + 9x - 10$ (C) $2x^2 + 13x - 10$ (D) $4x^2 + 9x - 10$

Show the answer

The answer is (A). The first product is $3x^2 + 15x - 2x - 10 = 3x^2 + 13x - 10$ , and subtracting $x(x - 4) = x^2 - 4x$ gives

\begin{aligned} &3x^2 + 13x - 10 - x^2 + 4x \\[0.4em] &= 2x^2 + 17x - 10. \end{aligned}

Choice (B) subtracts $4x$ instead of adding it, forgetting that the minus sign flips $-4x$ . Choice (C) drops the $x$ term of the second product entirely. Choice (D) adds $x(x - 4)$ instead of subtracting it.

If $(2x + k)(x - 3) = 2x^2 + 5x - 33$ for all values of $x$ , where $k$ is a constant, what is the value of $k$ ?

Show the answer

The answer is $11$ . The left side expands to $2x^2 + (k - 6)x - 3k$ . Matching constants gives $-3k = -33$ , so $k = 11$ , and the $x$ coefficient agrees since $11 - 6 = 5$ .

What is the degree of $(2x^3 - x)(x^2 + 5) - 2x^5$ ? (A) $2$ (B) $3$ (C) $5$ (D) $6$

Show the answer

The answer is (B). Expanding gives $2x^5 + 10x^3 - x^3 - 5x - 2x^5 = 9x^3 - 5x$ , and the $x^5$ terms cancel. Choice (A) counts the two terms of $9x^3 - 5x$ instead of reading the highest exponent. Choice (C) is the degree of the product before the subtraction. Choice (D) multiplies the degrees $3$ and $2$ instead of adding them.

Expanding and Combining Polynomials

Theory, part 43

Practice, part 44

Key terms

Polynomial

Degree

Like terms

Distributive property

Product of polynomials

Equivalent expressions

Coefficient matching

Check yourself

How this lesson connects

Builds on

Used later in