In the expression above, is a real constant. If the expression is equivalent to for all , which of the following could be the value of ?

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SAT Equivalent Expressions Question 228 | Free SAT Prep | Aniko

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Question 228·Hard·Equivalent Expressions

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\frac{x^2 + kx + 6}{x + 3}

In the expression above, $k$ is a real constant. If the expression is equivalent to $x + k - 3$ for all $x \ne -3$ , which of the following could be the value of $k$ ?

When an expression with a variable in the coefficients is said to be equivalent to another expression for all $x$ (except where undefined), clear any denominators so you get a polynomial identity. Expand both sides, then match coefficients of like powers of $x$ —often only one coefficient (like the constant term) gives a simple linear equation in the unknown parameter. Solving that small equation is much faster and more reliable than plugging in random $x$ -values or trying to do polynomial long division repeatedly for each answer choice.

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Eliminate the fraction

Start by getting rid of the denominator. Since the expression is equal to $x + k - 3$ for all $x \ne -3$ , what can you multiply both sides by so that the denominator disappears?

Relate the numerator to a product

After you clear the denominator, you should get an equation where $x^2 + kx + 6$ equals a product involving $(x+3)$ and $(x + k - 3)$ . Write that equation explicitly.

Expand and compare

Carefully expand $(x+3)(x + k - 3)$ and compare the result to $x^2 + kx + 6$ . Which parts (coefficients of $x^2$ , of $x$ , and the constant terms) must match?

Focus on the constant term

Notice that the $x^2$ and $x$ coefficients on both sides already match automatically. Use the constant terms to set up a simple linear equation in $k$ and solve it.

Desmos Guide

Enter the two expressions with a slider for k

In one expression line, type y = (x^2 + kx + 6)/(x + 3) and in another line type y = x + k - 3. When you type k, Desmos will prompt you to add a slider; create the slider for $k$ .

Test each answer choice for k

Use the slider (or type directly into the slider box) to set $k$ equal to each of the answer choices: 4, 5, 6, and 8. For each value, look at the graphs of the two expressions.

Check for complete overlap of the graphs

For the correct value of $k$ , the two graphs should lie exactly on top of each other for all $x$ -values except possibly at $x=-3$ , where the rational expression has a hole. For the incorrect values of $k$ , the two graphs will not coincide everywhere (they may intersect in a few points but will not be the same line).

Optional: Graph the difference to confirm

Add a third expression: y = (x^2 + kx + 6)/(x + 3) - (x + k - 3). For the correct value of $k$ , this graph will be the horizontal line $y=0$ (except possibly at $x=-3$ ). For other values of $k$ , the graph will not be identically zero.

Step-by-step Explanation

Clear the denominator

Because the two expressions are equal for all $x \ne -3$ , you can multiply both sides by $x+3$ (which is never zero on the allowed domain) to remove the denominator:

\frac{x^2 + kx + 6}{x+3} = x + k - 3

Multiply both sides by $x+3$ :

\frac{x^2 + kx + 6}{x+3}(x+3) = (x + k - 3)(x+3)

So the numerator must satisfy

x^2 + kx + 6 = (x+3)(x + k - 3).

Expand the product on the right-hand side

Now expand $(x+3)(x + k - 3)$ step by step.

First distribute $x$ :

(x+3)(x + k - 3) = x(x + k - 3) + 3(x + k - 3).

Then expand each term:

x(x + k - 3) = x^2 + x(k - 3),

3(x + k - 3) = 3x + 3k - 9.

Add them together:

(x+3)(x + k - 3) = x^2 + x(k - 3) + 3x + 3k - 9.

Combine the $x$ terms $x(k - 3) + 3x$ to get $kx$ :

(x+3)(x + k - 3) = x^2 + kx + 3k - 9.

Match the constant terms and solve for k

From Step 1 and the expansion in Step 2, you now have

x^2 + kx + 6 = x^2 + kx + 3k - 9.

Because these polynomials are equal for all $x$ , their constant terms must be equal:

6 = 3k - 9.

Solve for $k$ :

Add 9 to both sides:

6 + 9 = 3k

15 = 3k

Divide both sides by 3:

k = 5.

So the value of $k$ is 5, which corresponds to answer choice B.

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Step-by-step Explanation

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Step-by-step Explanation