[tex] \huge \boxed{\mathfrak{Question} \downarrow}[/tex]
Factorise the polynomials.
[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]
1. b² + 8b + 7
[tex] \sf \: b ^ { 2 } + 8 b + 7[/tex]
Factor the expression by grouping. First, the expression needs to be rewritten as b²+pb+qb+7. To find p and q, set up a system to be solved.
[tex] \sf \: p+q=8 \\ \sf pq=1\times 7=7 [/tex]
As pq is positive, p and q have the same sign. As p+q is positive, p and q are both positive. The only such pair is the system solution.
[tex] \sf \: p=1 \\ \sf \: q=7 [/tex]
Rewrite [tex]\sf\:b^{2}+8b+7 \: as \: \left(b^{2}+b\right)+\left(7b+7\right)[/tex].
[tex] \sf \: \left(b^{2}+b\right)+\left(7b+7\right) [/tex]
Take out the common factors.
[tex] \sf \: b\left(b+1\right)+7\left(b+1\right) [/tex]
Factor out common term b+1 by using distributive property.
[tex] \boxed{\boxed{ \bf \: \left(b+1\right)\left(b+7\right) }}[/tex]
__________________
2. 4x² + 4x + 1
[tex] \sf \: 4 x ^ { 2 } + 4 x + 1[/tex]
Factor the expression by grouping. First, the expression needs to be rewritten as 4x²+ax+bx+1. To find a and b, set up a system to be solved.
[tex] \sf \: a+b=4 \\ \sf ab=4\times 1=4 [/tex]
As ab is positive, a and b have the same sign. As a+b is positive, a and b are both positive. List all such integer pairs that give product 4.
[tex] \sf \: 1,4 \\ \sf 2,2 [/tex]
Calculate the sum for each pair.
[tex] \sf \: 1+4=5 \\ \sf \: 2+2=4 [/tex]
The solution is the pair that gives sum 4.
[tex] \sf \: a=2 \\ \sf b=2 [/tex]
Rewrite [tex]\sf4x^{2}+4x+1 as \left(4x^{2}+2x\right)+\left(2x+1\right)[/tex].
[tex] \sf \: \left(4x^{2}+2x\right)+\left(2x+1\right) [/tex]
Factor out 2x in 4x² + 2x.
[tex] \sf \: 2x\left(2x+1\right)+2x+1 [/tex]
Factor out common term 2x+1 by using distributive property.
[tex] \sf\left(2x+1\right)\left(2x+1\right) [/tex]
Rewrite as a binomial square.
[tex] \boxed{ \boxed{\bf\left(2x+1\right)^{2}} }[/tex]
__________________
3. 5n² + 10n + 20
[tex] \sf \: 5 n ^ { 2 } + 10 n + 20[/tex]
Factor out 5. Polynomial n² + 2n+4 is not factored as it does not have any rational roots.
[tex] \boxed{\boxed{\bf5\left(n^{2}+2n+4\right)} }[/tex]
__________________
4. m³ - 729
[tex] \sf \: m ^ { 3 } - 729[/tex]
Rewrite m³-729 as m³ - 9³. The difference of cubes can be factored using the rule: [tex]a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right)[/tex]. Polynomial m²+9m+81 is not factored as it does not have any rational roots.
[tex] \boxed{ \boxed{ \bf \: \left(m-9\right)\left(m^{2}+9m+81\right) }}[/tex]
__________________
5. x² - 81
[tex] \sf \: x ^ { 2 } - 81[/tex]
Rewrite x²-81 as x² - 9². The difference of squares can be factored using the rule: [tex] a^{2}-b^{2}=\left(a-b\right)\left(a+b\right)[/tex].
[tex] \boxed{ \boxed{ \bf\left(x-9\right)\left(x+9\right) }}[/tex]
__________________
6. 15x² - 17x - 4
[tex] \sf \: 15 x ^ { 2 } - 17 x - 4[/tex]
Factor the expression by grouping. First, the expression needs to be rewritten as 15x²+ax+bx-4. To find a and b, set up a system to be solved.
[tex] \sf \: a+b=-17 \\ \sf \: ab=15\left(-4\right)=-60 [/tex]
As ab is negative, a and b have the opposite signs. As a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -60.
[tex] \sf \: 1,-60 \\ \sf 2,-30 \\ \sf3,-20 \\ \sf4,-15 \\ \sf5,-12 \\ \sf 6,-10 [/tex]
Calculate the sum for each pair.
[tex] \sf \: 1-60=-59 \\ \sf 2-30=-28 \\ \sf3-20=-17 \\ \sf4-15=-11 \\ \sf5-12=-7 \\ \sf6-10=-4 [/tex]
The solution is the pair that gives sum -17.
[tex] \sf \: a=-20 \\ \sf \: b=3 [/tex]
Rewrite [tex]\sf15x^{2}-17x-4[/tex] as [tex]\sf \left(15x^{2}-20x\right)+\left(3x-4\right)[/tex].
[tex] \sf\left(15x^{2}-20x\right)+\left(3x-4\right) [/tex]
Factor out 5x in 15x²-20x.
[tex] \sf \: 5x\left(3x-4\right)+3x-4 [/tex]
Factor out common term 3x-4 by using distributive property.
[tex] \boxed{\boxed{ \bf\left(3x-4\right)\left(5x+1\right) }}[/tex]
