10. Multiplying Polynomials
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Learning Objective:
Multiply polynomials by applying the idea of partial products, i.e., adding together the products of each pair of terms in the polynomials. Multiply a polynomial with two terms and its conjugate, where the conjugate is another polynomial with the same two terms but the sign between the terms is changed. Expand and simplify expressions that involve multiplying and adding/subtracting polynomials.
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Notes:
Multiplying polynomials that each have two terms
Recall how to multiply the sum of two integers, a and b, by the sum of two integers, c and d: (a + b)(c + d) = ac + ad + bc + bd. We refer to these four partial products as FOIL, standing for “first, outer, inner, last.” For example, (3 + 4) × (2 + 1) = (3 × 2) + (3 × 1) + (4 × 2) + (4 × 1) = 6 + 3 + 8 + 4 = 21.
Let’s apply this idea to multiplying polynomials, 2x² + 3 and x – 4:
- (2x² + 3)(x – 4) = (2x²)(x) + (2x²)(-4) + (3)(x) + (3)(-4) = 2x³ – 8x² + 3x – 12.
More examples:
- (x² – 2x)(-3x + 1) = (x²)(-3x) + (x²)(1) + (-2x)(-3x) + (-2x)(1)
= -3x³ + x² + 6x² – 2x = -3x³ + 7x² – 2x. - (3x – 2)² = (3x – 2)(3x – 2) = (3x)(3x) + (3x)(-2) + (-2)(3x) + (-2)(-2)
= 9x² – 6x – 6x + 4 = 9x² – 12x + 4. Here, two of the partial products are combined. - (4x + 3)(4x – 3)
= (4x)(4x) + (4x)(-3) + (3)(4x) + (3)(-3)
= 16x² – 12x + 12x – 9 = 16x² – 9. Here, two of the partial products cancel out to zero. The term 4x – 3 is called the conjugate of 4x + 3.
Multiplying polynomials with more than two terms:
When there are more than two terms in the polynomials, we must make sure we remember to multiply each term in one polynomial by each term in the other polynomial. For example:
- (-2x³ – 3x + 4)(3x² – 1) = (-2x³)(3x²) + (-2x³)(-1) + (-3x)(3x²) + (-3x)(-1) + (4)(3x²) + (4)(-1)
= -6x⁵ + 2x³ – 9x³ + 3x + 12x² – 4
= -6x⁵ – 7x³ + 12x² + 3x – 4.
Polynomial multiplication and addition together:
- (-x + 3)(4x² + 3x) – 2x(-3x² + 2x + 4) = -4x³ – 3x² + 12x² + 9x + 6x³ – 4x² – 8x
= 2x³ + 5x² + x. - Check by plugging in the value x = 2: (12) – 4(-4) = 38 ⇒ 16 + 20 + 2 = 38 ⇒ 38 = 38.
More examples:
- (-4x³ + x)(2x² – 3) = (-4x³)(2x²) + (-4x³)(-3) + (x)(2x²) + (x)(-3)
= -8x⁵ + 12x³ + 2x³ – 3x = -8x⁵ + 14x³ – 3x. - -3x²(2x² – x – 4) + (2x² + 3x)(2x² – 3x) = -6x⁴ + 3x³ + 12x² + 4x³ – 9x²
= -2x⁴ + 3x³ + 3x². Note the conjugate in this example.
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