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Alexxandr [17]
3 years ago
14

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

Mathematics
1 answer:
Tasya [4]3 years ago
7 0

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

  • Factorise the polynomials.

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

__________________

<h3><u>1. x² + 4x + 4</u></h3>

{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} }}

__________________

<h3><u>2. x² - 8x + 16</u></h3>

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} }}

__________________

<h3><u>3. 4x² + 12xy + 9y²</u></h3>

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} }}

__________________

<h3><u>4. x⁴ - 2x² + 1</u></h3>

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}}}

__________________

  • <em>Refer to the attached picture too.</em>

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