19. Polynomial Operations

A polynomial is written in standard form with powers descending. In \(4x^3-2x^2+7x-5\), the degree is 3 and the leading coefficient is 4.

Add polynomials by combining matching powers:
\[ (3x^2+2x-1)+(x^2-5x+6)=4x^2-3x+5. \]

Subtract carefully by distributing the minus sign to every term:
\[ (5x^2+x-4)-(2x^2-3x+1)=3x^2+4x-5. \]

Multiply a monomial using distribution and exponent rules:
\[ 3x(2x^2-5x+4)=6x^3-15x^2+12x. \]

For two binomials, every term in the first factor multiplies every term in the second:
\[ (2x-3)(x+5)=2x^2+10x-3x-15=2x^2+7x-15. \]

Special patterns save time:
\[ (a+b)^2=a^2+2ab+b^2, \]
\[ (a-b)(a+b)=a^2-b^2. \]
So \((x+4)^2=x^2+8x+16\), not \(x^2+16\).

Geometry connection. If a rectangle has sides \(x+3\) and \(x+5\), its area is \((x+3)(x+5)=x^2+8x+15\). Algebraic multiplication describes how the smaller regions combine.

Common mistake: combining unlike powers, such as turning \(x^2+x\) into \(2x^3\). Addition does not change exponents.

Tip: after multiplying expressions of degrees \(m\) and \(n\), the product should usually have degree \(m+n\).

Enriched GED® problem-solving lab

Learning targets

• add and subtract polynomials by aligned powers
• multiply monomials and binomials distributively
• recognize and use special product patterns

Visual explanation

How to read this SVG. For binomial multiplication, every row label must multiply every column label. The four regions make missing products visible and show why the two middle terms are later combined. The same structure extends beyond the FOIL mnemonic.

Reliable problem-solving method

1. Write both polynomials in descending-power standard form.
2. Distribute a subtraction sign across the entire second polynomial.
3. For multiplication, record every pairwise product before combining.
4. Predict the product degree and verify with an alternate area or grid layout.

Worked example 1: Subtract polynomials

Problem. Simplify \((5x^2+x-4)-(2x^2-3x+1)\).

Solution.

1. Distribute the subtraction: \(5x^2+x-4-2x^2+3x-1\).
2. Combine squared and linear terms: \(3x^2+4x\).
3. Combine constants: \(-4-1=-5\).

Answer. \(3x^2+4x-5\)

Worked example 2: Multiply binomials

Problem. Expand \((2x-3)(x+5)\).

Solution.

1. Partial products: \(2x^2+10x-3x-15\).
2. Combine the middle terms: \(10x-3x=7x\).
3. The product is quadratic, as expected from degree 1 times degree 1.

Answer. \(2x^2+7x-15\)

Worked example 3: Square a binomial

Problem. Expand \((x+4)^2\).

Solution.

1. Rewrite as \((x+4)(x+4)\).
2. Multiply: \(x^2+4x+4x+16\).
3. Combine: \(x^2+8x+16\).

Answer. \(x^2+8x+16\)

Independent self-check

Mastery habit. Say what the answer means, include units when relevant, and verify it in the original condition.