➕ Algebra 1 · Exponents and Polynomials
Multiplying Polynomials practice questions
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Multiply (4x3)(5x2).
Answer: 20x5. Multiply the coefficients: 4 × 5 = 20. The variable is the same, so add the exponents: 3 + 2 = 5. The answer is 20x5. The choice 20x6 multiplies the exponents (3 × 2 = 6) instead of adding them, and 9x5 adds the coefficients (4 + 5 = 9) instead of multiplying them. The choice 9x6 makes both mistakes.
A student multiplies a binomial by a trinomial by writing the product of every term in the first factor with every term in the second. How many products does she write before combining like terms?
Answer: 6. Each of the 2 terms in the binomial multiplies each of the 3 terms in the trinomial, so there are 2 × 3 = 6 products. The choice 5 adds the term counts (2 + 3) instead of multiplying them. The choice 4 is the number of products in FOIL, which covers only two binomials (2 × 2). The choice 3 is just the number of terms in the trinomial.
Expand 2x2(3x − 4).
Answer: 6x3 − 8x2. Multiply 2x2 by each term. 2x2 × 3x = 6x3 (multiply the coefficients and add the exponents: 2 + 1 = 3). 2x2 × (−4) = −8x2. The answer is 6x3 − 8x2. The choice 6x3 + 8x2 loses the sign of the −4. The choice 6x3 − 4 skips multiplying the second term. The choice 6x3 − 8 forgets that the x2 stays with the 8.
Expand (x + 8)(x − 5).
Answer: x2 + 3x − 40. Multiply every term by every term: x × x = x2, x × (−5) = −5x, 8 × x = 8x and 8 × (−5) = −40. Combine the like terms: −5x + 8x = 3x. The answer is x2 + 3x − 40. The choice x2 + 13x − 40 treats the −5 as +5 in the middle. The choice x2 + 3x + 40 makes the last product positive, but a positive times a negative is negative. The choice x2 − 40 keeps only the first and last products.
Expand (3x − 2)(2x + 5).
Answer: 6x2 + 11x − 10. The four products are 3x × 2x = 6x2, 3x × 5 = 15x, (−2) × 2x = −4x and (−2) × 5 = −10. Combine the like terms: 15x − 4x = 11x. The answer is 6x2 + 11x − 10. The choice 6x2 + 19x − 10 treats −2 as +2 in the middle. The choice 5x2 + 11x − 10 adds the coefficients 3 + 2 instead of multiplying them. The choice 6x2 + 11x + 10 makes the last product positive.
Expand (x2 + 6)(x − 4).
Answer: x3 − 4x2 + 6x − 24. There are 2 × 2 = 4 products: x2 × x = x3, x2 × (−4) = −4x2, 6 × x = 6x and 6 × (−4) = −24. The powers x3, x2 and x and the constant are all different, so nothing combines. The choice ending in +24 gets the last sign wrong. The choice x3 − 4x2 − 24 drops the 6x product. The choice x3 + 2x2 − 24 wrongly merges −4x2 and 6x, which are not like terms.
Expand (x − 2)(x2 + 3x − 4).
Answer: x3 + x2 − 10x + 8. There are six products: x × x2 = x3, x × 3x = 3x2, x × (−4) = −4x, (−2) × x2 = −2x2, (−2) × 3x = −6x and (−2) × (−4) = +8. Combine the like terms: 3x2 − 2x2 = x2 and −4x − 6x = −10x. The answer is x3 + x2 − 10x + 8. The choice ending in −8 forgets that a negative times a negative is positive. The choice x3 + 5x2 + 2x − 8 treats −2 as +2. The choice x3 + x2 − 4x + 8 leaves out the product (−2)(3x) = −6x.
What is the degree of the product (2x3 − x + 1)(x2 + 5) after it is multiplied out and like terms are combined?
Answer: 5. The factors have degrees 3 and 2. For nonzero polynomials the degree of the product is the sum of the degrees: 3 + 2 = 5. You can see it in the highest-power product, 2x3 × x2 = 2x5, which nothing else can cancel. The full product is 2x5 + 9x3 + x2 − 5x + 5. The choice 6 multiplies the degrees (3 × 2). The choices 3 and 2 are the degrees of the factors themselves.
Dana expanded (x + 7)(x − 2) and got x2 − 14. Which test value would NOT show that her answer is wrong?
Answer: x = 0. The correct expansion is x2 + 5x − 14, so Dana left out the middle term 5x. Her answer can only look right at a value where 5x is 0, which is x = 0: both (0 + 7)(0 − 2) and 02 − 14 equal −14. At the other values the two sides differ. At x = 1 they are −8 and −13, at x = 3 they are 10 and −5, and at x = 4 they are 22 and 2.
Let x be a positive number of meters. A rectangular garden is x + 4 meters wide and 2x + 3 meters long. Which expression gives its area in square meters?
Answer: 2x2 + 11x + 12. Area is width times length: (x + 4)(2x + 3). The four products are x × 2x = 2x2, x × 3 = 3x, 4 × 2x = 8x and 4 × 3 = 12. Combine the like terms: 3x + 8x = 11x. The area is 2x2 + 11x + 12 square meters. The choice 2x2 + 12 keeps only the first and last products. The choice 2x2 + 11x + 7 adds the constants (4 + 3) instead of multiplying them. The choice 3x + 7 adds the width and length instead of multiplying them.