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Question:
Consider the expression [tex]\frac{x^2 + x -10}{x + 3}[/tex]
When using the inspection method the number you would add to (and subtract from) the constant term of the numerator so the polynomial in the numerator will have (x + 3) as a factor is?
Answer:
The constant to add is 4
Step-by-step explanation:
Given
[tex]\frac{x^2 + x -10}{x + 3}[/tex]
First, we need to get an expression that has x + 3 has its factor.
Represent this expression with: [tex](x + 3)(x + k)[/tex]
Expand
[tex]x^2 + 3x + kx + 3k[/tex]
Group like terms
[tex]x^2 + (3 + k)x + 3k[/tex]
Compare the above expression to: [tex]x^2 + x - 10[/tex]
[tex](3 + k)x = x[/tex]
[tex]3k = -10[/tex]
However, we only consider solving [tex](3 + k)x = x[/tex] for k
[tex](3 + k)x = x[/tex]
[tex]3 + k = 1[/tex]
Subtract 3 from both sides
[tex]3 - 3 + k = 1 - 3[/tex]
[tex]k = 1 - 3[/tex]
[tex]k= -2[/tex]
Substitute -2 for k in [tex](x + 3)(x + k)[/tex]
[tex](x + 3)(x + k) = (x + 3)(x -2)[/tex]
[tex](x + 3)(x + k) = x^2 + 3x - 2x - 6[/tex]
[tex](x + 3)(x + k) = x^2 + x - 6[/tex]
So, the expression that has a factor of x + 3 is [tex]x^2 + x - 6[/tex]
To get the constant term to add/subtract, we have:
[tex]Constant = (x^2 + x - 6) - (x^2 + x - 10)[/tex]
Open brackets
[tex]Constant = x^2 + x - 6 - x^2 - x + 10[/tex]
Collect Like Terms
[tex]Constant = x^2 - x^2+ x - x - 6+ 10[/tex]
[tex]Constant = - 6+ 10[/tex]
[tex]Constant = 4[/tex]