Question

How do you multiply out \[\left( {8k - 3} \right)\left( {{k^2} - k + 1} \right)\]?

Answer 1

Hint: To solve this question, we have to expand the bracket. This expansion of brackets means multiplying everything inside the bracket by the letter or number outside the bracket. Here, we are given two brackets, therefore we need to multiply both the terms in the first bracket and each term of the second bracket.

Complete step by step solution:
We are given \[\left( {8k - 3} \right)\left( {{k^2} - k + 1} \right)\]. We will now multiply both the terms in the first bracket with the second bracket.
\[\left( {8k - 3} \right)\left( {{k^2} - k + 1} \right)\]
\[ \Rightarrow 8k\left( {{k^2} - k + 1} \right) - 3\left( {{k^2} - k + 1} \right)\]
Now we will multiply each term individually.
For the first term:
Multiplication of $ 8k $ and $ {k^2} $ is: $ 8k \times {k^2} = 8{k^{1 + 2}} = 8{k^3} $ .
Multiplication of $ 8k $ and $ - k $ is: $ 8k \times \left( { - k} \right) = - 8{k^{1 + 1}} = - 8{k^2} $ .
Multiplication of $ 8k $ and $ 1 $ is: $ 8k \times 1 = 8k $ .
For the second term:
Multiplication of $ - 3 $ and $ {k^2} $ is: $ - 3 \times {k^2} = - 3{k^2} $ .
Multiplication of $ - 3 $ and $ - k $ i s: $ - 3 \times \left( { - k} \right) = 3k $ .
Multiplication of $ - 3 $ and $ 1 $ is: $ - 3 \times 1 = - 3 $ .
Thus we can write
\[\left( {8k - 3} \right)\left( {{k^2} - k + 1} \right)\]
\[
   \Rightarrow 8k\left( {{k^2} - k + 1} \right) - 3\left( {{k^2} - k + 1} \right) \\
   \Rightarrow 8{k^3} - 8{k^2} + 8k - 3{k^2} + 3k - 3 \\
 \]
Now. We will rearrange it by combining like terms together.
We know that there are two pairs of like terms: $ - 8{k^2} $ and $ - 3{k^2} $ , $ 8k $ and $ 3k $ .
Thus, we can write:
\[
   \Rightarrow 8{k^3} - 8{k^2} - 3{k^2} + 8k + 3k - 3 \\
   \Rightarrow 8{k^3} - 11{k^2} + 11k - 3 \\
 \]

Thus, by multiplying out \[\left( {8k - 3} \right)\left( {{k^2} - k + 1} \right)\], we get \[8{k^3} - 11{k^2} + 11k - 3\] as our final answer.

Note: While solving this question, we have used the concept of combining the like terms. These like terms can be defined as the terms with the same variable and sane power of variable. The coefficients of the terms can be different. For example, in our case, $ - 8{k^2} $ and $ - 3{k^2} $ , $ 8k $ and $ 3k $ are the two like [airs which we have combined to get our final answer.