➕ Algebra 1 · Exponents and Polynomials
Special Products of Binomials practice questions
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Expand (x + 9)2.
Answer: x2 + 18x + 81. Square the first term, double the product of the two terms, and square the last term: x2 + 2(x)(9) + 92 = x2 + 18x + 81. The choice x2 + 81 leaves out the middle term (it squares each part separately). The choice x2 + 9x + 81 forgets to double. The choice x2 + 18x + 18 doubles the 9 instead of squaring it.
Expand (y − 8)2.
Answer: y2 − 16y + 64. Here a = y and b = 8, and the binomial has a minus sign. Square the first term: y2. Double the product, keeping the minus sign: −2(y)(8) = −16y. Square the last term: 82 = +64, because a negative times a negative is positive. The answer is y2 − 16y + 64. The choice y2 − 64 is what (y + 8)(y − 8) gives. The choice y2 − 16y − 64 makes the last term negative. The choice y2 + 16y + 64 drops the minus sign in the middle.
Expand (4x + 5)2.
Answer: 16x2 + 40x + 25. Here a = 4x and b = 5. Square the first term, including the 4: (4x)2 = 16x2. Double the product: 2(4x)(5) = 40x. Square the last term: 25. The answer is 16x2 + 40x + 25. The choice 4x2 + 40x + 25 squares only the x. The choice 16x2 + 20x + 25 forgets to double. The choice 16x2 + 25 leaves out the middle term.
Expand (2x + 3)(2x − 3).
Answer: 4x2 − 9. These use the same two terms, 2x and 3, with a plus in one factor and a minus in the other, so the middle terms cancel: (2x)2 − 32 = 4x2 − 9. The choice 2x2 − 9 forgets to square the 2. The choice 4x2 + 9 puts a plus between the squares. The choice 4x2 − 12x + 9 is (2x − 3)2, the square of a difference.
Use a special product to find 48 × 52 in your head. What is the product?
Answer: 2496. Both numbers are 2 away from 50: 48 × 52 = (50 − 2)(50 + 2) = 502 − 22 = 2500 − 4 = 2496. The choice 2504 adds the 4 instead of subtracting it. The choice 2498 subtracts 2 instead of 22 = 4. The choice 2500 forgets the correction altogether.
What is the value of (a + b)2 − (a2 + b2), in terms of a and b?
Answer: 2ab. Expand: (a + b)2 = a2 + 2ab + b2. Subtract a2 + b2, and what is left is the middle term, 2ab. For example, with a = 2 and b = 5 the difference is 49 − 29 = 20 = 2 × 2 × 5. The choice 0 is the classic mistake of thinking the two expressions are equal.
Exactly one of these products has a middle term after it is multiplied out. Which one?
Answer: (x + 5)(x − 4). The last three each use the same two terms with a plus in one factor and a minus in the other, so their middle terms cancel: they equal x2 − 100, 9x2 − 1 and x4 − 4. In (x + 5)(x − 4) the numbers 5 and 4 are different, so the middle terms −4x and +5x do not cancel. It equals x2 + x − 20.
Use the identity (a + b)2 − (a − b)2 = 4ab to find 172 − 132.
Answer: 120. Match a + b = 17 and a − b = 13. Adding these gives 2a = 30, so a = 15, and then b = 2. So 172 − 132 = 4ab = 4 × 15 × 2 = 120. Check directly: 289 − 169 = 120. The choice 16 squares the difference 17 − 13. The choice 60 uses 2ab instead of 4ab. The choice 884 uses 17 and 13 themselves as a and b (4 × 17 × 13).
A square garden has side x meters. A path 2 meters wide runs all the way around it, so the garden and path together form a larger square with side x + 4 meters. How many square meters does the path alone cover?
Answer: 8x + 16. The path is the big square minus the garden. The big square has area (x + 4)2 = x2 + 8x + 16 (square the first term, double the product 2 × x × 4 = 8x, square the last term). The garden has area x2. Subtract: x2 + 8x + 16 − x2 = 8x + 16 square meters. The choice 16 comes from the mistake (x + 4)2 = x2 + 16. The choice 4x + 16 counts only two of the four side strips. The choice x2 + 8x + 16 is the area of the whole big square, path plus garden.
Expand (x2 − 5)2.
Answer: x4 − 10x2 + 25. Here a = x2 and b = 5. Square the first term: (x2)2 = x4 (multiply the exponents). Double the product, keeping the minus sign: −2(x2)(5) = −10x2. Square the last term: 25. The answer is x4 − 10x2 + 25. The choice x4 − 25 is what (x2 + 5)(x2 − 5) gives. The choice x4 − 10x + 25 forgets that x2 stays in the middle term. The choice x4 + 10x2 + 25 drops the minus sign.