Question

How do you find the product of ${\left( {9 - 7x} \right)^2}$?

Answer 1

Hint: In this question we have to find the product of the given expression which is in form of ${\left( {a + bx} \right)^2}$, convert the expression as the product of two binomials, i.e., ${\left( {a + bx} \right)^2} = \left( {a + bx} \right)\left( {a + bx} \right)$, then by using the FOIL method which is a technique used to help remember the steps required to multiply two binomials. Multiply each term in the first binomial with each term in the second binomial using Foil method as shown,
$\left( {ax + b} \right)\left( {cx + d} \right) = ax \cdot cx + ax \cdot d + b \cdot cx + b \cdot d$, now by substituting the values in the formula we will get the required product.

Complete step by step solution:
The FOIL method is a technique used to help remember the steps required to multiply two binomials. Remember that when you multiply two terms together you must multiply the coefficient (numbers) and add the exponents, and finally combine the like terms.
Now given expression is ${\left( {9 - 7x} \right)^2}$,
This is in form of ${\left( {a + bx} \right)^2}$, which can be rewritten as ${\left( {a + bx} \right)^2} = \left( {a + bx} \right)\left( {a + bx} \right)$,
Now rewriting the given expression, we get,
$ \Rightarrow {\left( {9 - 7x} \right)^2} = \left( {9 - 7x} \right)\left( {9 - 7x} \right)$,
Now using Foil method, Multiply the first terms of both the binomials that are $9$ from $9 - 7x$ and $9$from $9 - 7x$, The product of $9$and $9$i.e.,
$9 \cdot 9 = 81$,
Now then we will multiply the outer terms of both the binomials, the product of outer terms that are $9$ from $9 - 7x$and $ - 7x$ from $9 - 7x$i.e.,
$9 \cdot - 7x = - 63x$,
Now multiply the inner terms of the binomials. The inner terms here are $ - 7x$ from $9 - 7x$ and $9$ from $9 - 7x$i.e.,
$ - 7x \cdot 9 = - 63x$,
 At last multiply the last terms in each of the two binomials, the last two terms here $ - 7x$ from $9 - 7x$and $ - 7x$ from $9 - 7x$, so the product will be i.e.,
$ - 7x \cdot - 7x = 49{x^2}$,
So this is can represented as,
$ \Rightarrow {\left( {9 - 7x} \right)^2} = \left( {9 - 7x} \right)\left( {9 - 7x} \right) = 9 \cdot 9 + 9 \cdot \left( { - 7x} \right) + \left( { - 7x} \right) \cdot 9 + \left( { - 7x} \right)\left( { - 7x} \right)$,
By simplifying we get,
$ \Rightarrow {\left( {9 - 7x} \right)^2} = 81 - 63x - 63x + 49{x^2}$,
Now by combining the like terms we get,
$ \Rightarrow {\left( {9 - 7x} \right)^2} = 81 - 126x + 49{x^2}$
Final Answer:
$\therefore $The product of the expression ${\left( {9 - 7x} \right)^2}$ will be equal to $81 - 126x + 49{x^2}$.

Note: Steps of foil method will be: First multiply the first terms, then the outer terms, then the inner terms and finally the last terms.
The product of two positive will be positive.
The product of two negatives will also be positive.
The product of a positive and negative will always be negative.