Directions: Determine whether each trinomial is a perfect square. If so, factor it.
2. 4a^2+4a+1
Answer: 2(a+1)^2
3. 9m^2+15m+25
Answer: no
4. d^2-22d+121
Answer: (d-11)^2
Directions: Determine whether each binomial is the difference of squares. If so, factor it.
5. x^2-16
Answer: (x-4)(x+4)
6. y^2-20
Answer: no
7. 16m^2-25n^2
Answer: (m+4)-(5+n)
8. 8a^2-18
Answer: (7+ab)-(7-ab)
#2, no
all you had to do is expand your answer to see that it can't be right.
should have been (2a + 1)^2
#7, your answer is not in factored form, it shows a subtraction, not a multiplication.
just look at your answer!
how can (m+4)-(5+n) possible be 16m^2-25n^2 ????
should have been (4m+5n)(4m-5n)
in #8, first take out a "common factor" of 2, then you should see the difference of squares
8a^2-18
= 2(4a^2 - 9) = 2(2a+3)(2a-3)
again, how could your answer of
(7+ab)-(7-ab) possibly turn into 8a^2 - 18 ??? And where did the b come from ???
To determine whether a trinomial is a perfect square, you need to consider the coefficients and the exponents of the variables.
To factor a perfect square trinomial, follow these steps:
1. Check if the coefficient of the squared term and the constant term are perfect squares.
2. Take the square root of the squared term's coefficient.
3. Write two sets of parentheses.
4. Put the square root of the squared term's coefficient as the first term in each set of parentheses.
5. Write the variable(s) with half of the coefficient of the linear term.
6. Write the variable(s) again, but with the opposite sign compared to the linear term.
Let's apply these steps to the given trinomials:
2. 4a^2 + 4a + 1
The coefficient of the squared term is 4, which is a perfect square (2^2). The constant term is 1, also a perfect square (1^2).
We take the square root of 4, which is 2.
The factored form is: 2(a + 1)^2.
3. 9m^2 + 15m + 25
The coefficient of the squared term is 9, which is a perfect square (3^2). However, the constant term is 25, not a perfect square.
Therefore, this trinomial is not a perfect square.
4. d^2 - 22d + 121
The coefficient of the squared term is 1, which is a perfect square (1^2). The constant term is 121, also a perfect square (11^2).
We take the square root of 1, which is 1.
The factored form is: (d - 11)^2.
To determine whether a binomial is the difference of squares, you need to check if it follows the pattern (a^2 - b^2), where a and b are terms involving variables.
To factor the difference of squares, follow these steps:
1. Identify if the binomial follows the pattern (a^2 - b^2).
2. Write two sets of parentheses.
3. Put the square root of the first term as the first term in each set of parentheses.
4. Write the variable(s) with the same signs as in the original binomial.
5. Put the square root of the second term as the second term in each set of parentheses, but with opposite signs compared to the original binomial.
Let's apply these steps to the given binomials:
5. x^2 - 16
This binomial follows the pattern (a^2 - b^2), where a = x and b = 4.
The factored form is: (x - 4)(x + 4).
6. y^2 - 20
This binomial does not follow the difference of squares pattern, as there is no second term to take the square root of.
Therefore, this binomial cannot be factored using the difference of squares method.
7. 16m^2 - 25n^2
This binomial follows the pattern (a^2 - b^2), where a = 4m and b = 5n.
The factored form is: (4m + 5n)(4m - 5n).
8. 8a^2 - 18
This binomial does not follow the difference of squares pattern, as there is a non-zero constant term.
Therefore, this binomial cannot be factored using the difference of squares method.