Video: Writing Rational Expressions in Different Forms by Comparing Coefficients

Write 4𝑥³ − 2𝑥² + 7 in the form (𝑎𝑥² + 𝑏𝑥 + 𝑐)(𝑥 − 5) + 𝑑 by comparing coefficients.

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Video Transcript

Write four 𝑥 cubed minus two 𝑥 squared plus seven in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 multiplied by 𝑥 minus five plus 𝑑 by comparing coefficients.

In order to answer this question, we need to write the expression four 𝑥 cubed minus two 𝑥 squared plus seven in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 multiplied by 𝑥 minus five plus 𝑑. We will do this by firstly working out the values of 𝑎, 𝑏, and 𝑐. Our first step will be to expand the two brackets or parentheses. In order to do this, we need to multiply each term in the first bracket by each of the terms in the second bracket.

𝑎𝑥 squared multiplied by 𝑥 is equal to 𝑎𝑥 cubed. Multiplying 𝑎𝑥 squared by negative five gives us negative five 𝑎𝑥 squared. We then need to multiply 𝑏𝑥 by 𝑥 minus five. 𝑏𝑥 multiplied by 𝑥 is 𝑏𝑥 squared. And, 𝑏𝑥 multiplied by negative five is negative five 𝑏𝑥. Finally, we multiply 𝑐 by 𝑥 minus five. 𝑐 multiplied by 𝑥 is equal to 𝑐𝑥. And, 𝑐 multiplied by negative five is negative five 𝑐. We also need to drop the 𝑑 down to the next line.

At this stage, we have one term with 𝑥 cubed, two terms with 𝑥 squared, two terms with 𝑥, and two constant terms. We can now work out the values of 𝑎, 𝑏, 𝑐, and 𝑑 by comparing coefficients. The coefficient of 𝑥 cubed on the left-hand side of the equation is four. The coefficient of 𝑥 cubed on the right-hand side is 𝑎. Therefore, four is equal to 𝑎 or 𝑎 equals four.

When we consider the 𝑥 squared terms, we have negative two on the left-hand side and negative five 𝑎 plus 𝑏 on the right-hand side. This means that negative two is equal to negative five 𝑎 plus 𝑏. As 𝑎 is equal to four, this can be rewritten as negative two is equal to negative five multiplied by four plus 𝑏. Negative five multiplied by four is equal to negative 20. Adding 20 to both sides of this equation gives us 18 is equal to 𝑏. This is because negative two plus 20 equals 18.

There is no 𝑥 term on the left-hand side of the equation. On the right-hand side, we have negative five 𝑏𝑥 and positive 𝑐𝑥. Comparing the coefficients gives us zero is equal to negative five 𝑏 plus 𝑐. As 𝑏 is equal to 18, this can be rewritten as zero equals negative five multiplied by 18 plus 𝑐. Negative five multiplied by 18 is equal to negative 90. Adding 90 to both sides of this equation gives us 90 is equal to 𝑐.

The constant term on the left-hand side of the equation is seven. And on the right-hand side, we have negative five 𝑐 plus 𝑑. This means that seven is equal to negative five 𝑐 plus 𝑑. 𝑐 is equal to 90, so we can substitute this into the equation. Negative five multiplied by 90 is equal to negative 450, as negative five multiplied by nine is equal to negative 45. We can add 450 to both sides of this equation to work out the value of 𝑑. 𝑑 is equal to 457.

We now have values for 𝑎, 𝑏, 𝑐, and 𝑑. 𝑎 is equal to four. 𝑏 is equal to 18. 𝑐 is equal to 90. And, 𝑑 is equal to 457. The expression four 𝑥 cubed minus two 𝑥 squared plus seven can be rewritten as four 𝑥 squared plus 18𝑥 plus 90 multiplied by 𝑥 minus five plus 457. We could check our answer by expanding the parentheses and simplifying. This would give us the initial expression, four 𝑥 cubed minus two 𝑥 squared plus seven.

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