Lesson 3 · from §44 of the 1910 text
Binomials with a Common Term
Multiplying x + a by x + b: (x + a)(x + b) = x^2 + (a + b)x + ab.
IV. The product of two binomials with a common term equals the square of the common term, plus the sum of the unlike terms times the common term, plus the product of the unlike terms.
So (x + 3)(x - 5) = x^2 + (3 - 5)x + (3)(-5) = x^2 - 2x - 15.
Quick check
(y + 6)(y - 2)
Backward, to split x^2 + 7x + 12 into two binomials, look for two numbers whose product is 12 and whose sum is 7: 3 and 4. So x^2 + 7x + 12 = (x + 3)(x + 4).
Quick check
Write x^2 + 9x + 18 as the product of two binomials.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Expand by the formula.
(x + 1)(x + 2)
(x + 2)(x + 3)
(x + 3)(x + 4)
(x + 4)(x + 5)
(x + 5)(x + 6)
(n + 1)(n - 2)
(n - 2)(n + 3)
(n - 3)(n + 4)
(n - 4)(n + 5)
(n - 5)(n + 6)
(a - 1)(a - 2)
(a - 2)(a - 3)
(a - 3)(a - 4)
(a - 4)(a - 5)
(a - 4)(a - 6)
(2y + 3)(2y + 4)
(2y + 2)(2y + 3)
(3a + 1)(3a + 4)
(2n + 3)(2n + 5)
(2n + 3)(2n - 5)
(2n + 3)(2n - 4)
(2a + 1)(2a - 5)
(3a - 5)(3a + 1)
(4a + 3)(4a - 5)
(4a - 5)(4a - 6)
(4ab + 1)(4ab - 6)
(3x - 2)(3x + 5)
(4a - 3b)(4a + 5b)
(5a - 6b)(5a - 7b)
Divide in your head.
\dfrac{x^2 + 3x + 2}{x + 1}
\dfrac{x^2 + 5x + 6}{x + 2}
\dfrac{x^2 + 4x + 3}{x + 3}
\dfrac{x^2 + 6x + 5}{x + 1}
\dfrac{x^2 + 7x + 12}{x + 3}
\dfrac{x^2 - 7x + 12}{x - 3}
\dfrac{x^2 - 5x + 6}{x - 3}
\dfrac{x^2 - 5x + 4}{x - 1}
\dfrac{x^2 - 6x + 5}{x - 5}
\dfrac{x^2 - 7x + 6}{x - 6}
\dfrac{x^2 - 7x + 12}{x - 4}
Find an exact binomial divisor of each.
x^2 + 3x + 2
x^2 + 5x + 6
x^2 + 7x + 12
x^2 + 6x + 8
x^2 + 7x + 10
x^2 + 8x + 15
x^2 - 3x + 2
x^2 - 5x + 6
x^2 - 6x + 8
x^2 - 8x + 15
x^2 - 8x + 12
x^2 - 9x + 14
Write each as the product of two binomials.
x^2 + 8x + 7
x^2 + 9x + 8
x^2 - 10x + 9
x^2 - 11x + 10
x^2 - 10x + 16
x^2 + 10x + 21
x^2 + 9x + 20
x^2 - 9x + 18
x^2 + 9x + 14
x^2 - 12x + 32
x^2 + 11x + 10