➕ Algebra 1 · Exponents and Polynomials
Multiplying Polynomials: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
(−2x3)(8x5) = −16x8.
✓ True. Multiply the coefficients: −2 × 8 = −16. Add the exponents of the same variable: 3 + 5 = 8.
7x2 × 2x3 = 14x6.
✗ False. The exponents add: 2 + 3 = 5, so the product is 14x5. Multiplying the exponents, 2 × 3 = 6, is the mistake.
Multiplying a binomial by a trinomial gives six products before like terms are combined.
✓ True. Each of the 2 terms in the binomial multiplies each of the 3 terms in the trinomial: 2 × 3 = 6.
FOIL gives the correct set of products when you multiply any two polynomials.
✗ False. FOIL describes four products, which is right only for two binomials. A binomial times a trinomial has six products, and FOIL would miss two of them.
(x − 2)(x − 9) = x2 − 11x + 18.
✓ True. The middle term is −9x − 2x = −11x, and (−2)(−9) = +18. Check with x = 4: (2)(−5) = −10 and 16 − 44 + 18 = −10.
(x + 7)(x − 3) = x2 − 21.
✗ False. That keeps only the first and last products. The middle term is −3x + 7x = 4x, so the answer is x2 + 4x − 21. Check with x = 4: 11 × 1 = 11, but 16 − 21 = −5.
The product of two polynomials is always another polynomial.
✓ True. Multiplying monomials only adds whole-number exponents, so every term of the product is still a monomial. Polynomials are closed under multiplication.
A degree-4 polynomial times a degree-1 polynomial has degree 4.
✗ False. The degrees add for nonzero polynomials, so the product has degree 4 + 1 = 5. For example, x4 × x = x5.
A nonzero degree-3 polynomial times a nonzero degree-4 polynomial has degree 7.
✓ True. The highest-power term of the product is the product of the two highest-power terms, like x3 × x4 = x7, and nothing can cancel it.
(x + 10)2 = x2 + 100.
✗ False. The exponent does not distribute over a sum. (x + 10)2 = (x + 10)(x + 10) = x2 + 20x + 100. Check with x = 3: 132 = 169, but 32 + 100 = 109.
x × x3 = x4.
✓ True. A variable with no written exponent has exponent 1, so x1 × x3 = x1+3 = x4.
Testing an expansion with x = 0 will catch a missing x-term.
✗ False. At x = 0 every term that contains x is 0, so a missing x-term makes no difference. For example, (x + 6)(x + 2) and the wrong answer x2 + 12 both equal 12 at x = 0.
If both sides agree at x = 3, that proves the expansion is correct for every value of x.
✗ False. A wrong answer can match by luck at one value. For example, x2 + 11 and (x + 1)(x + 2) both equal 20 at x = 3, but at x = 4 they give 27 and 30.