Some students are asked to write down linear and quadratic expressions that have $(x + 2)$ as a factor. (a) The expressions $3x + 5$, $x + 1$, and $2x - 10$ are examples of linear expressions in $x$. Write down a linear expression in $x$, other than $x + 2$, that has $x + 2$ as a factor. (b) Anton writes down a quadratic expression of the form $x^2 - k$, where $k$ is a number. For what value of $k$ will Anton's expression have $x + 2$ as a factor? (c) To get her quadratic expression, Denise multiplies $x + 2$ by $2x + 3$. Find Denise's expression. $ (x + 2)(2x + 3) = $ (d) Fiona's expression is $3x^2 + 11x + 10$. She uses division to check if $x + 2$ is a factor of it. Explain how division will allow her to check this. (ii) Divide $3x^2 + 11x + 10$ by $x + 2$. (e) Write down one quadratic expression, other than those already given above, that has $x + 2$ as a factor.

Junior Cycle Mathematics ALGEBRA - Factorising: Some students are asked to write

Some students are asked to write down linear and quadratic expressions that have $(x + 2)$ as a factor - Junior Cycle Mathematics - Question 13 - 2014

Question 13

Some students are asked to write down linear and quadratic expressions that have $(x + 2)$ as a factor. (a) The expressions $3x + 5$, $x + 1$, and $2x - 10$ are exa... show full transcript

Worked Solution & Example Answer:Some students are asked to write down linear and quadratic expressions that have $(x + 2)$ as a factor - Junior Cycle Mathematics - Question 13 - 2014

Step 1

Write down a linear expression in $x$, other than $x + 2$, that has $x + 2$ as a factor.

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Answer

One possible linear expression is $2x + 4 = 2(x + 2)$ , which includes $x + 2$ as a factor.

Step 2

For what value of $k$ will Anton's expression have $x + 2$ as a factor?

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In the expression $x^2 - k$ , to have $x + 2$ as a factor, we can rewrite it using the difference of squares:

oot1{4})(x + oot1{4})$$ Thus, we find that $k = 4$.

Step 3

Find Denise's expression.

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Using the multiplication, we have:

$(x + 2)(2x + 3) = x(2x + 3) + 2(2x + 3)$ $= 2x^2 + 3x + 4x + 6$ $= 2x^2 + 7x + 6$

So, Denise's expression is $2x^2 + 7x + 6$ .

Step 4

Explain how division will allow her to check if $x + 2$ is a factor.

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If $x + 2$ is a factor of the polynomial $3x^2 + 11x + 10$ , the polynomial should divide evenly by $x + 2$ , resulting in no remainder. If the remainder is 0, then $x + 2$ is indeed a factor.

Step 5

Divide $3x^2 + 11x + 10$ by $x + 2$.

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Performing synthetic or polynomial division gives:

Divide the leading term: $3x^2 / x = 3x$ .
Multiply back: $3x(x + 2) = 3x^2 + 6x$ .
Subtract: $(3x^2 + 11x + 10) - (3x^2 + 6x) = 5x + 10$ .
Next, divide $5x + 10$ by $x + 2$ : $5$ . The result is $3x + 5$ with a remainder of $0$ . Therefore, $3x^2 + 11x + 10 = (x + 2)(3x + 5)$ .

Step 6

Write down one quadratic expression, other than those already given above, that has $x + 2$ as a factor.

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Answer

A quadratic expression that has $x + 2$ as a factor could be obtained by multiplying $x + 2$ by another linear expression, for example,

$(x + 2)(x + 1) = x^2 + 3x + 2$ .

This gives the quadratic $x^2 + 3x + 2$ , which has $x + 2$ as a factor.

Some students are asked to write down linear and quadratic expressions that have $(x + 2)$ as a factor - Junior Cycle Mathematics - Question 13 - 2014

Worked Solution & Example Answer:Some students are asked to write down linear and quadratic expressions that have $(x + 2)$ as a factor - Junior Cycle Mathematics - Question 13 - 2014

Write down a linear expression in $x$, other than $x + 2$, that has $x + 2$ as a factor.

For what value of $k$ will Anton's expression have $x + 2$ as a factor?

Find Denise's expression.

Explain how division will allow her to check if $x + 2$ is a factor.

Divide $3x^2 + 11x + 10$ by $x + 2$.

Write down one quadratic expression, other than those already given above, that has $x + 2$ as a factor.

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