8.1 Polynomial Operations and Structure

8.1 Polynomial Operations and Structure

College Board’s Next-Generation ACCUPLACER Advanced Algebra and Functions (AAF) test weights polynomial equations at 5–15% of the 20-item computer-adaptive exam — typically 1–3 items. Table 11 names three skills: creating polynomial equations in one and two variables, solving them, and graphing polynomial functions. This section is the algebra those later skills assume: degree, leading coefficient, standard form, and the operations that build a polynomial from pieces. You cannot create or graph a cubic if you cannot expand a binomial times a trinomial.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

Degree, leading coefficient, and standard form

A polynomial in x is a finite sum of terms of the form a_k x^k where each exponent k is a nonnegative integer. No square roots of x, no x in a denominator, and no exponential 2^x. The degree is the highest power of x that actually appears after like terms are combined. The leading term is that highest-power term. The leading coefficient is the number multiplying it. Standard form writes the terms in descending powers of x.

A constant nonzero polynomial such as −4 has degree 0. Linear polynomials have degree 1, quadratics degree 2, cubics degree 3, quartics degree 4. Degree is not the number of terms: x^5 + 1 is degree 5 with only two terms. AAF will also bait you with a cancelled leading term. Combine first: x^3 + 4x − x^3 + 2 is 4x + 2, degree 1, not 3.

When you multiply polynomials, degrees add. A degree-1 factor times a degree-2 factor is degree 3. When you add polynomials, the degree is the larger of the two degrees unless the leading terms cancel. When you subtract, the same rule applies after every sign in the second polynomial has flipped.

Naming from a product without fully expanding

(2x − 1)(x^2 + 5) is already a product of degree 1 and degree 2, so the product is degree 3. The leading term is 2x · x^2 = 2x^3, so the leading coefficient is 2. You can answer a “what is the degree / leading coefficient” item without writing every middle term — but you still need the full expansion when the stem asks for the equivalent polynomial in standard form.

Adding and subtracting

To add, write both polynomials in standard form and combine like powers. To subtract, distribute the minus across every term of the second polynomial. The last term is the usual casualty.

Worked addition: (3x^3 − 5x + 2) + (x^2 + 5x − 7)

Line the powers up mentally: there is a 3x^3 from the first polynomial, an x^2 from the second, then the linear terms −5x + 5x, then the constants 2 − 7.

3x^3 − 5x + 2 + x^2 + 5x − 7 = 3x^3 + x^2 + (−5x + 5x) + (2 − 7)

= 3x^3 + x^2 − 5

The x terms cancel. The degree of the sum is 3, matching the first summand, and the leading coefficient is 3. Cancellation of a leading term would drop the degree; cancellation of a middle term, as here, does not.

Worked subtraction: (4x^2 − 3x + 1) − (x^2 − 8x + 6)

Distribute the minus: 4x^2 − 3x + 1 − x^2 + 8x − 6.

Combine: (4 − 1)x^2 + (−3 + 8)x + (1 − 6) = 3x^2 + 5x − 5.

The trap 4x^2 − 3x + 1 − x^2 − 8x + 6 forgets to flip the last two signs of the subtrahend and produces 3x^2 − 11x + 7, a different quadratic. Subtracting a polynomial is adding its opposite: every sign inside the second set of parentheses changes, including a minus that is already there (−(−8x) becomes +8x).

Multiplying: distribution, FOIL, and binomial times trinomial

Distribution is the only multiplication rule: each term of the first factor multiplies each term of the second, then like terms combine. FOIL (First, Outer, Inner, Last) is distribution when both factors are binomials. A binomial times a trinomial is six products, not four. FOIL is the wrong acronym there, but the same distributive law applies.

Worked example: expand (x + 4)(x^2 − 2x + 5)

Distribute x across the trinomial, then distribute 4:

x(x^2 − 2x + 5) = x^3 − 2x^2 + 5x

4(x^2 − 2x + 5) = 4x^2 − 8x + 20

Add the two rows:

x^3 + (−2x^2 + 4x^2) + (5x − 8x) + 20 = x^3 + 2x^2 − 3x + 20

Result: (x + 4)(x^2 − 2x + 5) = x^3 + 2x^2 − 3x + 20.

Check three cheap diagnostics before you move on:

• Degree: 1 + 2 = 3. The expanded form is cubic.
• Leading term: x · x^2 = x^3. Leading coefficient 1.
• Constant term: 4 · 5 = 20.
• Numeric check: at x = 1, the product is (5)(1 − 2 + 5) = 5 · 4 = 20, and 1 + 2 − 3 + 20 = 20. Match.

A common incomplete expansion keeps x^3 − 2x^2 + 5x and forgets to add 4x^2 − 8x + 20. Another writes x^3 − 2x^2 + 5x + 4 as if the 4 were a constant addend rather than a factor. Parentheses mean multiply. A third trap copies only the outer and last products and reports x^3 + 20, which is missing every middle term.

Two binomials: FOIL as a special case

(2x − 3)(x + 5) = 2x · x + 2x · 5 + (−3) · x + (−3) · 5 = 2x^2 + 10x − 3x − 15 = 2x^2 + 7x − 15.

First rebuilds 2x^2, Outer and Inner combine to the middle term, Last rebuilds −15. If the middle coefficient is wrong but the ends look right, you dropped a sign on Outer or Inner. (2x − 3)(x + 5) is not 2x^2 − 15; that dropped-middle answer is the same family of error as writing (x + 5)^2 = x^2 + 25.

Special products, including squaring a binomial

Memorize the templates. They are faster than six-term distribution, and they are exactly what AAF tests when the stem says “expand” or “which expression is equivalent.”

Worked: (x + 5)^2

(x + 5)^2 = x^2 + 2 · x · 5 + 25 = x^2 + 10x + 25.

The trap x^2 + 25 is (x)^2 + (5)^2, which is not the square of the binomial. The middle term 2ab is required. Check with a number: if x = 1, (6)^2 = 36, and 1 + 10 + 25 = 36, while 1 + 25 = 26. It is the same one identity, over and over, including when a is already a monomial such as 2x.

Worked: (2x − 3)^2

Here a = 2x and b = 3. The minus template gives a^2 − 2ab + b^2:

(2x)^2 − 2(2x)(3) + 9 = 4x^2 − 12x + 9.

Trap: 4x^2 − 9 treats it as a difference of squares. Trap: 4x^2 + 9 drops the middle and the minus. Trap: 4x^2 − 6x + 9 uses 2 · 3 instead of 2 · (2x) · 3. Square the whole first term: (2x)^2 = 4x^2, not 2x^2.

Worked: (x + 3)(x − 3)

Difference of squares: x^2 − 9. No middle term. Expanding FOIL also gives x^2 − 3x + 3x − 9 = x^2 − 9. The linear terms cancel; that cancellation is the identity. Do not “complete” it by inventing a middle 6x.

Squaring is not the cube identity. (x + 2)^3 = x^3 + 6x^2 + 12x + 8, whereas x^3 + 8 is a sum of cubes that factors as (x + 2)(x^2 − 2x + 4). Do not swap those. The factoring versions of these identities live in GCF, Grouping, and Special Products; here you are running them forward, from factors to an expanded polynomial.

Creating polynomial equations in one and two variables

Table 11 starts with creating, not only simplifying. A one-variable polynomial equation often comes from a product, a volume, or consecutive-integer language. A two-variable polynomial is a function y = P(x) you write from a description or from roots plus a point.

One variable — consecutive integers. Three consecutive integers have product 60 more than the middle integer. If the middle integer is n, then:

(n − 1) · n · (n + 1) = n + 60

The left side is n(n^2 − 1) = n^3 − n. Subtract n + 60 from both sides:

n^3 − n − n − 60 = 0

n^3 − 2n − 60 = 0

That is a cubic equation in one variable. You will solve it in Solving Polynomial Equations; here the skill is writing it. Do not skip the “60 more than the middle” clause and set the product equal to 60. The created equation still includes the = n + 60 relationship until you move every term to one side.

One variable — box. A rectangular box has a square base of side s inches and height 4 inches more than the side. Its volume is 96 cubic inches:

s^2 (s + 4) = 96

s^3 + 4s^2 − 96 = 0

Created form can stay factored (s^2(s + 4) = 96) until a stem asks for standard form. Expanding is Section 8.1; setting the result equal to zero is the handoff to Section 8.2.

Two variables — from a verbal rule. “y is 3 times the cube of x, minus twice x, plus 7” is the unique cubic

y = 3x^3 − 2x + 7.

Two variables — from factors plus a point. If a cubic has roots x = −1, x = 2, and x = 5, then

y = a(x + 1)(x − 2)(x − 5).

The extra point determines a. If the graph passes through (0, 20):

20 = a(1)(−2)(−5) = 10a, so a = 2.

y = 2(x + 1)(x − 2)(x − 5).

Expand only if the options are in standard form: y = 2(x + 1)(x^2 − 7x + 10) = 2(x^3 − 7x^2 + 10x + x^2 − 7x + 10) = 2(x^3 − 6x^2 + 3x + 10) = 2x^3 − 12x^2 + 6x + 20. Creating in two variables is the same skill graphing will reverse in Graphing Polynomials, Roots, and End Behavior: factors encode roots; a encodes stretch and the sign of the end behavior.

AAF operation checklist

1. Combine like terms, then name degree and leading coefficient from standard form.
2. Add by lining up powers. Subtract by flipping every sign in the second polynomial.
3. Multiply by distributing every term of the first factor through the second. Degrees add; constants multiply.
4. For (a ± b)^2, write a^2 ± 2ab + b^2. Never drop 2ab.
5. To create an equation, name the variable, translate each phrase, and do not skip the “equals” statement. Move everything to one side only when you are ready to solve.
6. Check an expansion by degree, by the constant term, and by substituting one easy x-value.

Polynomial operations are the cheap points inside the 5–15% polynomial block. The CAT will not give a formula sheet. If expanding (x + 4)(x^2 − 2x + 5) is automatic, solving and graphing in the next two sections sit on a firm base. Handheld calculators are not allowed on AAF except with an approved accommodation (College Board calculator policy); distribution is a pencil skill.