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Chapter 8: Algebraic Expressions and Identities

    Introduction

    Addition and Subtraction of Algebraic Expressions

    Multiplication of Algebraic Expressions: Introduction

    Multiplying a Monomial by a Monomial

    Multiplying a Monomial by a Polynomial

    Multiplying a Polynomial by a Polynomial

    Exercise 8.1

    Exercise 8.2

    Exercise 8.3

    Exercise 8.4

    What Have We Discussed ?

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Chapter 8: Algebraic Expressions and Identities > Multiplying a Polynomial by a Polynomial

Multiplying a Polynomial by a Polynomial

Multiplying a binomial by a binomial

Let us multiply one binomial (2a + 3b) by another binomial, say (3a + 4b). We do this step-by-step, as we did in earlier cases, following the distributive law of multiplication.

Instructions

Expand the expression

  • The expression is (3a + 4b) × (2a + 3b) then we divide the terms
  • multiply the terms with factors
  • Hence multiply the values we get a2 + ab + ba + b2
  • (Since ba = ab ) Add the terms 6a2+ ab + 12b2
  • We have found the answer.

Observe, every term in one binomial multiplies every term in the other binomial.

When we carry out term by term multiplication, we expect 2 × 2 = terms to be present. But two of these are like terms, which are combined, and hence we get 3 terms.

In multiplication of polynomials with polynomials, we should always look for like terms, if any, and combine them.

Example 8: Multiply

(i) (x – 4) and (2x + 3)

Instructions

(x – 4) × (2x + 3) = x × (2x + 3) – 4 × (2x + 3)
(x × 2x) + (x × 3) – (4 × 2x) – (4 × 3)
+ + +
Adding like terms: – –

(ii) (x – y) and (3x + 5y)

(x – y) × (3x + 5y) = x(3x+5y) − y(3x+5y)
= + − −
= 3x2 + − 5y2

Example 9: Multiply

Instructions

(i) (a + 7) and (b – 5)

(a + 7) × (b – 5) = a(b−5) + 7(b−5)
= + + −

(ii)(a2 + 2b2) and (5a – 3b)

(a2 + 2b2) x (5a – 3b) = a2(5a – 3b) + 2b2 × (5a – 3b)
= - + -

Multiplying a binomial by a trinomial

In this multiplication, we shall have to multiply each of the three terms in the trinomial by each of the two terms in the binomial. We shall get in all 3 × 2 = terms, which may reduce to 5 or less, if the term by term multiplication results in like terms.

a+7 × a2+3a+5 = a × a2+3a+5 + 7 × a2+3a+5 = + + + + +

= a3 + (3a2 + 7a2 ) + ( 5a + 21a ) + 35

= a3 + + + 35

Why are there only 4 terms in the final result?

Example 10: Simplify (a + b) (2a – 3b + c) – (2a – 3b) c

Instructions

Expand the expression

  • Exapanding the expression and multiply with a,b and c terms (a + b) (2a – 3b + c) – (2a – 3b) c
  • separate the each terms
  • (Note, –3ab and 2ab are like terms)Now add the terms are a2 - - - b2 +bc
  • We have found the answer.