This topic gives an overview of;
Look at the following patterns of dots.
Think of similar other situations in which two algebraic expressions have to be multiplied. Ameena gets up. She says, “We can think of area of a rectangle.” The area of a rectangle is l × b, where l is the length, and b is breadth. If the length of the rectangle is increased by 5 units, i.e., (l + 5) and breadth is decreased by 3 units , i.e., (b – 3) units, the area of the new rectangle will be (l + 5) × (b – 3).
Think about volume(The volume of a rectangular box is given by the product of its length, breadth and height).
Sarita points out that when we buy things, we have to carry out multiplication. For example, if
price of bananas per dozen = Rs p
and for the school picnic bananas needed = z dozens,
then we have to pay = Rs p × z
Suppose, the price per dozen was less by Rs 2 and the bananas needed were less by 4 dozens.
Then, price of bananas per dozen = Rs (p – 2)
and bananas needed = (z – 4) dozens,
Therefore, we would have to pay = Rs (p – 2) × (z – 4)
In all the above examples, we had to carry out multiplication of two or more quantities. If the quantities are given by algebraic expressions, we need to find their product. This means that we should know how to obtain this product. Let us do this systematically. To begin with we shall look at the multiplication of two monomials.
We begin with 4 × x = x + x + x + x = 4x as seen earlier. Similarly, 4 × (3x) = 3x + 3x + 3x + 3x = 12x.
Now, observe the following products.
(i) x × 3y = x × 3 × y = 3 × x × y = 3xy
(ii) 5x × 3y = 5 × x × 3 × y = 5 × 3 × x × y = 15xy
(iii) 5x × (–3y) = 5 × x × (–3) × y = 5 × (–3) × x × y = –15xy
(iv)5x × 4x2 = (5 × 4) × (x × x2) = 20 × x3 = 20x3
(v) 5x × (– 4xyz) = (5 × – 4) × (x × xyz) = –20 × (x × x × yz) = –20x2yz
Observe how we collect the powers of different variables in the algebraic parts of the two monomials. While doing so, we use the rules of exponents and powers.
Observe the following examples.
(i) 2x × 5y × 7z = (2x × 5y) × 7z = 10xy × 7z = 70xyz
(ii) 4xy × 5x2y2 × 6x3y3 = (4xy × 5x2y2) × 6x3y3 = 20x3y3 × 6x3y3 = 120x3y3 × x3y3 = 120 (x3 × x3) × (y3 × y3) = 120x6 × y6 = 120x6y6
It is clear that we first multiply the first two monomials and then multiply the resulting monomial by the third monomial. This method can be extended to the product of any number of monomials.
Let us multiply the monomial 3x by the binomial 5y + 2, i.e., find 3x × (5y + 2) = ? Recall that 3x and (5y + 2) represent numbers.
Therefore, using the distributive law,
3x × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x
We commonly use distributive law in our calculations. For example:
7 × 106 = 7 × (100 + 6) = 7 × 100 + 7 × 6 (Here, we used distributive law)
= 700 + 42 = 742⇒ 7 × 38 = 7 × (40 – 2)
= 7 × 40 – 7 × 2 (Here, we used distributive law)
= 280 – 14 = 266
Consider 3p × (4p2 + 5p + 7). As in the earlier case, we use distributive law;
3p × (4p2 + 5p + 7) = (3p × 4p2) + (3p × 5p) + (3p × 7) = 12p3 + 15p2 + 21p. Multiply each term of the trinomial by the monomial and add products.
By using the distributive law, we are able to carry out themultiplication term by term.
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,
(3a + 4b) × (2a + 3b) = 3a × (2a + 3b) + 4b × (2a + 3b)
= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b)
= 6a2+ 17ab + 12b2(Since ba = ab)
When we carry out term by term multiplication, we expect 2 × 2 = 4 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.
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 = 6 terms, which may reduce to 5 or less, if the term by term multiplication results in like terms.
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