Mathematics, 18.09.2021 05:40 Lolag4996

Can you please help me give the Complete solution

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Can you please help me give the Complete solution (nonsence report )

Answers: 3

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Answer from: mikehager4321

$\huge \boxed{\mathfrak{Question} \downarrow}$

Factorise the polynomials.

$\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}$

__________________

1. x² + 4x + 4

${x}^{2} + 4x + 4$

Factor the expression by grouping. First, the expression needs to be rewritten as x²+ax+bx+4. To find a and b, set up a system to be solved.

$a+b=4 \\ ab=1\times 4=4$

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.

$1,4 \\ 2,2$

Calculate the sum for each pair.

$1+4=5 \\ 2+2=4$

The solution is the pair that gives sum 4.

$a=2 \\ b=2$

Rewrite x² + 4x + 4 as (x² + 2x) + (2x + 4)

$\left(x^{2}+2x\right)+\left(2x+4\right)$

Take out the common factors.

$x\left(x+2\right)+2\left(x+2\right)$

Factor out common term x+2 by using distributive property.

$\left(x+2\right)\left(x+2\right)$

Rewrite as a binomial square.

$b. \: \: \boxed{ \boxed{{(x + 2)}^{2} }}$

__________________

2. x² - 8x + 16

$x ^ { 2 } - 8 x + 16$

Factor the expression by grouping. First, the expression needs to be rewritten as x²+ax+bx+16. To find a and b, set up a system to be solved.

$a+b=-8 \\ ab=1\times 16=16$

As ab is positive, a and b have the same sign. As a+b is negative, a and b are both negative. List all such integer pairs that give product 16.

$-1,-16 \\ -2,-8 \\ -4,-4$

Calculate the sum for each pair.

$-1-16=-17 \\ -2-8=-10 \\ -4-4=-8$

The solution is the pair that gives sum -8.

$a=-4 \\ b=-4$

Rewrite x²-8x+16 as $\left(x^{2}-4x\right)+\left(-4x+16\right)$ .

$\left(x^{2}-4x\right)+\left(-4x+16\right)$

Take out the common factors.

$x\left(x-4\right)-4\left(x-4\right)$

Factor out common term x-4 by using distributive property.

$\left(x-4\right)\left(x-4\right)$

Rewrite as a binomial square.

$d. \: \: \boxed{\boxed{\left(x-4\right)^{2} }}$

__________________

3. 4x² + 12xy + 9y²

$4 x ^ { 2 } + 12 x y + 9 y ^ { 2 }$

Use the perfect square formula, $a^{2}+2ab+b^{2}=\left(a+b\right)^{2}$ , where a=2x and b=3y.

$e. \: \: \boxed{ \boxed{\left(2x+3y\right)^{2} }}$

__________________

4. x⁴ - 2x² + 1

$x ^ { 4 } - 2 x ^ { 2 } + 1$

To factor the expression, solve the equation where it equals to 0.

$x^{4}-2x^{2}+1=0$

By Rational Root Theorem, all rational roots of a polynomial are in the form p/q, where p divides the constant term 1 and q divides the leading coefficient 1. List all candidates p/q.

$± \: 1$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

x=1

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x⁴-2x²+1 by x-1 to get x³+x²-x-1. To factor the result, solve the equation where it equals to 0.

$x^{3}+x^{2}-x-1=0$

By Rational Root Theorem, all rational roots of a polynomial are in the form p/q, where p divides the constant term -1 and q divides the leading coefficient 1. List all candidates p/q.

$± \: \: 1$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

x=1

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x³+x²-x-1 by x-1 to get x²+2x+1. To factor the result, solve the equation where it equals to 0.

$x^{2}+2x+1=0$

All equations of the form ax²+bx+c=0 can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$ . Substitute 1 for a, 2 for b and 1 for c in the quadratic formula.

$x=\frac{-2±\sqrt{2^{2}-4\times 1\times 1}}{2} \\$

Do the calculations.

$x=\frac{-2±0}{2} \\$

Solutions are the same.

x=-1

Rewrite the factored expression using the obtained roots.

$\left(x-1\right)^{2}\left(x+1\right)^{2} \\ = a. \: \: \boxed{ \boxed{\left(x^{2}-1\right)^{2}}}$

__________________

Refer to the attached picture too.
Can you please help me give the Complete solution (nonsence report )

Answer from: Quest

step-by-step explanation:

Answer from: Quest

i believe the correct answer is option 4.principalities were small territories that were run by russian rulers.

which statement correctly defines principalities in northern asia in the 1200s? principalities were small territories that were run by russian rulers.

hope i could ! : )

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