Answer:
- x(x -3) = x² -3x
- (x +2)(x -2) = x² -4
Explanation:
When a polynomial has a zero at x=p, it means that the binomial (x -p) is a factor of that polynomial. When we know all the zeros of a polynomial, we can use this fact to write the polynomial in factored form.
The factored form is converted to standard form by multiplying out the factors. (The notion is similar for numbers: 2×3 is the factored form of the standard-form number 6, for example. 6 is what you get when you multiply 2 by 3.)
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1.
The zeros are x=0 and x=3, so the factors are (x -0)(x -3). This simplifies to ...
x(x -3)
When you carry out the multiplication using the distributive property, you get the standard form expression ...
x² -3x
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2.
The zeros are x=-2 and x=2, so the factors are (x -(-2))(x -2). This simplifies to ...
(x +2)(x -2)
When you carry out the multiplication using the distributive property, you get ...
x(x -2) +2(x -2) = x² -2x +2x -4
= x² -4
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Additional comment
You may have noticed that the x-terms cancelled in the second problem, so that there is no x-term in the standard form. This happens whenever you multiply the factors (x -a)(x +a) for any value of 'a'. This product is called a "special product" for that reason. It has many applications where cancelling the x-term is desired. It is a useful form to remember.
x² -a² = (x -a)(x +a) . . . . . difference of squares special product