- Factorise the polynomials.
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<h3><u>1. x² + 4x + 4</u></h3>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.
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.
Calculate the sum for each pair.
The solution is the pair that gives sum 4.
Rewrite x² + 4x + 4 as (x² + 2x) + (2x + 4)
Take out the common factors.
Factor out common term x+2 by using distributive property.
Rewrite as a binomial square.
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<h3><u>2. x² - 8x + 16</u></h3>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.
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.
Calculate the sum for each pair.
The solution is the pair that gives sum -8.
Rewrite x²-8x+16 as .
Take out the common factors.
Factor out common term x-4 by using distributive property.
Rewrite as a binomial square.
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<h3><u>3. 4x² + 12xy + 9y²</u></h3>Use the perfect square formula, , where a=2x and b=3y.
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<h3><u>4. x⁴ - 2x² + 1</u></h3>To factor the expression, solve the equation where it equals to 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.
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.
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.
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.
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.
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.
All equations of the form ax²+bx+c=0 can be solved using the quadratic formula: . Substitute 1 for a, 2 for b and 1 for c in the quadratic formula.
Do the calculations.
Solutions are the same.
Rewrite the factored expression using the obtained roots.
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- <em>Refer to the attached picture too.</em>
