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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.
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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.
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We will do this by firstly working out the values of 𝑎, 𝑏, and 𝑐.
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Our first step will be to expand the two brackets or parentheses.
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In order to do this, we need to multiply each term in the first bracket by each of the terms in the second bracket.
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𝑎𝑥 squared multiplied by 𝑥 is equal to 𝑎𝑥 cubed.
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Multiplying 𝑎𝑥 squared by negative five gives us negative five 𝑎𝑥 squared.
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We then need to multiply 𝑏𝑥 by 𝑥 minus five.
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𝑏𝑥 multiplied by 𝑥 is 𝑏𝑥 squared.
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And, 𝑏𝑥 multiplied by negative five is negative five 𝑏𝑥.
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Finally, we multiply 𝑐 by 𝑥 minus five.
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𝑐 multiplied by 𝑥 is equal to 𝑐𝑥.
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And, 𝑐 multiplied by negative five is negative five 𝑐.
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We also need to drop the 𝑑 down to the next line.
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At this stage, we have one term with 𝑥 cubed, two terms with 𝑥 squared, two terms with 𝑥, and two constant terms.
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We can now work out the values of 𝑎, 𝑏, 𝑐, and 𝑑 by comparing coefficients.
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The coefficient of 𝑥 cubed on the left-hand side of the equation is four.
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The coefficient of 𝑥 cubed on the right-hand side is 𝑎.
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Therefore, four is equal to 𝑎 or 𝑎 equals four.
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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.
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This means that negative two is equal to negative five 𝑎 plus 𝑏.
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As 𝑎 is equal to four, this can be rewritten as negative two is equal to negative five multiplied by four plus 𝑏.
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Negative five multiplied by four is equal to negative 20.
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Adding 20 to both sides of this equation gives us 18 is equal to 𝑏.
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This is because negative two plus 20 equals 18.
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There is no 𝑥 term on the left-hand side of the equation.
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On the right-hand side, we have negative five 𝑏𝑥 and positive 𝑐𝑥.
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Comparing the coefficients gives us zero is equal to negative five 𝑏 plus 𝑐.
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As 𝑏 is equal to 18, this can be rewritten as zero equals negative five multiplied by 18 plus 𝑐.
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Negative five multiplied by 18 is equal to negative 90.
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Adding 90 to both sides of this equation gives us 90 is equal to 𝑐.
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The constant term on the left-hand side of the equation is seven.
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And on the right-hand side, we have negative five 𝑐 plus 𝑑.
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This means that seven is equal to negative five 𝑐 plus 𝑑.
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𝑐 is equal to 90, so we can substitute this into the equation.
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Negative five multiplied by 90 is equal to negative 450, as negative five multiplied by nine is equal to negative 45.
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We can add 450 to both sides of this equation to work out the value of 𝑑.
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𝑑 is equal to 457.
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We now have values for 𝑎, 𝑏, 𝑐, and 𝑑.
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𝑎 is equal to four.
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𝑏 is equal to 18.
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𝑐 is equal to 90.
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And, 𝑑 is equal to 457.
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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.
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We could check our answer by expanding the parentheses and simplifying.
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This would give us the initial expression, four 𝑥 cubed minus two 𝑥 squared plus seven.