Factor the following polynomials
9-x^2
4x^2+81-36
4y2^+16y+16
x2^-25/x+5
assistance needed
i need assistance yes
hints
first one is a basic example of "difference of squares"
second does not factor
third is a perfect square, (2y+4)^2
the last one, factor the top as a difference of squares, then cancel one of those factors with the denominator.
what do u need help on
To factor the given polynomials, we need to express them as a product of simpler polynomials or binomials. Let's go through each polynomial one by one.
1. Polynomial: 9 - x^2
To factor this polynomial, we can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). Comparing it with our given polynomial, we can see that a = 3 and b = x. Therefore, we can rewrite the polynomial as (3 + x)(3 - x).
Factorization: (3 + x)(3 - x)
2. Polynomial: 4x^2 + 81 - 36
This polynomial has three terms. We can combine the constants together and then look for any common factors to factor out.
Combining the constants: 81 - 36 = 45
Now, we have 4x^2 + 45. We can see that there are no common factors we can factor out. Therefore, this polynomial cannot be factored any further.
Factorization: 4x^2 + 45
3. Polynomial: 4y^2 + 16y + 16
This polynomial also has three terms. We can again look for any common factors to factor out.
The constants 16 and 16 have a common factor of 4. Factoring out 4 leaves us with:
4(y^2 + 4y + 4)
Now, we can see that the polynomial inside the parentheses can be factored further. It is a perfect square trinomial, which is of the form a^2 + 2ab + b^2 = (a + b)^2.
In our case, a = y and b = 2. Therefore, we can rewrite the polynomial as:
4(y + 2)^2
Factorization: 4(y + 2)^2
4. Polynomial: (x^2 - 25) / (x + 5)
This polynomial is a fraction, but we can still factor the numerator.
The numerator (x^2 - 25) is a difference of squares, which we can factor using the same formula as mentioned earlier. a = x and b = 5.
(x + 5)(x - 5)
Now, we can rewrite the original polynomial as:
(x + 5)(x - 5) / (x + 5)
Here, we can see that (x + 5) appears in both the numerator and denominator. We can cancel out this common factor:
(x - 5)
Factorization: (x - 5)
Therefore, the factored form of the given polynomials are:
1. 9 - x^2 = (3 + x)(3 - x)
2. 4x^2 + 81 - 36 = 4x^2 + 45
3. 4y^2 + 16y + 16 = 4(y + 2)^2
4. (x^2 - 25) / (x + 5) = (x - 5)