Factor Difference of Two Squares (Harder) 3

Question 1 of 4

Factor.

$5u^4-5$

1. $5(u^2+1)(u-1)(u+1)$
2. $(5u^2+1)(u-1)(u+1)$
3. $5(u^2+1)(u+1)$
4. $5(u^2+1)(2u+1)$

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Factorising the Difference of Two Squares

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2=(\color{#00880A}{a}-\color{#9a00c7}{b})(\color{#00880A}{a}+\color{#9a00c7}{b})$

First, find the Greatest Common Factor (GCF) of the two terms.

Start by listing down their factors.

Factors of

$5u^4$ : $5$

$timesutimesutimesutimesu$

Factors of

$5$ :

$1times$ $5$

Both

$5u^4$ and

$5$ have

$5$ as their factor, so it is the GCF.

Next, factor by placing

$5$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by

$5$ , then simplify.

		$5[(5u^4div5)-(5div5)]$
	$=$	$5(u^4-1)$

Next, express both terms inside the parenthesis as perfect squares. In other words, both terms should have

$2$ as their exponent.

	$u^4-1$
$=$	$(u^2)^2-1$	$(u^2)^2=u^4$
$=$	$(u^2)^2-1^2$	$1^2=1$

Next, label the values in the expression

$(u^2)^2-1^2$ and substitute the values into the formula given for Factoring the Difference of Two Squares.

$a=u^2$

$b=2$

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$	$=$	$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})$
$5(\color{#00880A}{u^2})^2-\color{#9a00c7}{1}^2$	$=$	$5(\color{#00880A}{u^2}+\color{#9a00c7}{1})(\color{#00880A}{u^2}-\color{#9a00c7}{1})$

Now, express both terms inside the second parenthesis as perfect squares. In other words, both terms should have

$2$ as their exponent.

		$u^2-1$
	$=$	$u^2-1^2$	$1^2=1$

Finally, label the values in the expression

$u^2-1^2$ and substitute the values into the formula given for Factoring the Difference of Two Squares.

$a=u$

$b=1$

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$	$=$	$(\color{#00880A}{a}-\color{#9a00c7}{b})(\color{#00880A}{a}+\color{#9a00c7}{b})$
$5(u^2+1)(\color{#00880A}{u}^2-\color{#9a00c7}{1}^2)$	$=$	$5(u^2+1)(\color{#00880A}{u}-\color{#9a00c7}{1})(\color{#00880A}{u}+\color{#9a00c7}{1})$

$5(u^2+1)(u-1)(u+1)$

Question 2 of 4

Factor.

$(x+y)^2-z^2$

1. $(x+z)(y-z)$
2. $(x+y+z)(x+y-z)$
3. $(x-y+z)(x+y-z)$
4. $(x-y-z)(x+y+z)$

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Factoring the Difference of Two Squares

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2=(\color{#00880A}{a}-\color{#9a00c7}{b})(\color{#00880A}{a}+\color{#9a00c7}{b})$

Label the values in the expression

$(x+y)^2-z^2$ and substitute the values into the formula given for Factoring the Difference of Two Squares.

$a=x+y$

$b=z$

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$	$=$	$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})$
$(\color{#00880A}{x+y})^2-\color{#9a00c7}{z}^2$	$=$	$[(\color{#00880A}{x+y})+\color{#9a00c7}{z}][(\color{#00880A}{x+y})-\color{#9a00c7}{z}]$
	$=$	$(x+y+z)(x+y-z)$

$(x+y+z)(x+y-z)$

Question 3 of 4

Factor.

$y^2-(m-n)^2$

1. $(y-n)(y+m)$
2. $(y+m-n)(y-m-n)$
3. $(y+m-n)(y-m+n)$
4. $(y+mn)(y-mn)$

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Factoring the Difference of Two Squares

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2=(\color{#00880A}{a}-\color{#9a00c7}{b})(\color{#00880A}{a}+\color{#9a00c7}{b})$

Label the values in the expression

$y^2-(m-n)^2$ and substitute the values into the formula given for Factoring the Difference of Two Squares.

$a=y$

$b=m-n$

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$	$=$	$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})$
$\color{#00880A}{y}^2-(\color{#9a00c7}{m-n})^2$	$=$	$[\color{#00880A}{y}+(\color{#9a00c7}{m-n})][\color{#00880A}{y}-(\color{#9a00c7}{m-n})]$
	$=$	$(y+m-n)(y-m+n)$

$(y+m-n)(y-m+n)$

Question 4 of 4

factor.

$4a^2-4(b-c)^2$

1. $4(a-c)(a+b)$
2. $4(a+b)(a+c)$
3. $4(a-b+c)(b+c)$
4. $4(a+b-c)(a-b+c)$

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Factoring the Difference of Two Squares

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2=(\color{#00880A}{a}-\color{#9a00c7}{b})(\color{#00880A}{a}+\color{#9a00c7}{b})$

First, find the Greatest Common Factor (GCF) of the two terms.

Start by listing down their factors.

Factors of

$4a^2$ : $4$

$timesatimesa$

Factors of

$4(b-c)^2$ : $4$

$times(b-c)times(b-c)$

Both

$4a^2$ and

$4(b-c)^2$ have

$4$ as their factor, so it is the GCF.

Next, factor by placing

$4$ outside a bracket.

Also, place the given polynomial inside the bracket with each term divided by

$4$ , then simplify.

		$4[(4a^2div4)-(4(b-c)^2div4)]$
	$=$	$4(a^2-(b-c)^2)$

Since both terms have

$2$ as their exponents, they are perfect squares.

Factor the expression further by using the formula given for Factor the Difference of Two Squares.

$a=a$

$b=(b-c)$

$\color{#00880A}{a}^2-\color{#9a00c7}{b}^2$	$=$	$(\color{#00880A}{a}+\color{#9a00c7}{b})(\color{#00880A}{a}-\color{#9a00c7}{b})$
$4(\color{#00880A}{a}^2-\color{#9a00c7}{(b-c)}^2)$	$=$	$4(\color{#00880A}{a}+\color{#9a00c7}{(b-c)})(\color{#00880A}{a}-\color{#9a00c7}{(b-c)})$
	$=$	$4(a+b-c)(a-b+c)$	Remove the brackets inside

$4(a+b-c)(a-b+c)$

Factor Difference of Two Squares (Harder) 3

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Quiz summary

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1. Question

Hint

Fantastic!

Factorising the Difference of Two Squares

2. Question

Hint

Correct!

Factoring the Difference of Two Squares

3. Question

Hint

Well Done!

Factoring the Difference of Two Squares

4. Question

Hint

Great Work!

Factoring the Difference of Two Squares

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