factorize the following using.
formula #1 : (x+y)^3 = x^3+3x^2y+3xy^2+y^2
formula #2 : (x-y)^3 = x^3-3x^2y+3xy^2-y^2
Problem #1 : 8a^3+60a^2+150a+125
Problem #2 : 27x^3-27x^2y+9xy^2-y^3
please tell me how to solve them
actually in the end of both formulas its y^3 sorry....
What if I rewrote your first problem this way
8a^3+60a^2+150a+125
= (2a)^3 + 3(2a)^2 (5) + 3(2a)(5^2) + 5^3 ?
can you see the pattern?
do the same with the second problem,
Post your answers please
To factorize the given expressions, we will apply the formulas you provided.
Problem #1: 8a^3 + 60a^2 + 150a + 125
Step 1: Observe that the first and last terms are perfect cubes (8 = 2^3 and 125 = 5^3). Also, the middle terms are multiples of each other (60 = 3*20 = 3*2^2*5).
Step 2: Rewrite the expression and try to find a common factor that can be pulled out. In this case, the common factor is 5.
8a^3 + 60a^2 + 150a + 125
= 2^3 * a^3 + 3 * 2^2 * 5 * a^2 + 5 * 2 * 3 * 5 * a + 5^3
= 2^3 * a^3 + 3 * 2^2 * 5 * a^2 + 3 * 5^2 * a + 5^3
Step 3: Apply formula #1: (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3. Compare the expression obtained in step 2 with the pattern of formula #1.
2^3 * a^3 + 3 * 2^2 * 5 * a^2 + 3 * 5^2 * a + 5^3
= (2a + 5)^3
Therefore, the factorized form of 8a^3 + 60a^2 + 150a + 125 is (2a + 5)^3.
Problem #2: 27x^3 - 27x^2y + 9xy^2 - y^3
Step 1: In this case, none of the terms are perfect cubes. However, we can see that the entire expression is the difference of cubes since it follows the pattern A^3 - B^3.
Step 2: Rearrange the expression as a difference of cubes: (3x)^3 - (y)^3
Step 3: Apply formula #2: (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3. In this case, the pattern is reversed from the given expression. Therefore, we need to rewrite the expression as:
27x^3 - y^3 + (- 27x^2y + 9xy^2)
Step 4: Observe that the terms in parentheses resemble the pattern of formula #2. Factorize the expression inside parentheses:
27x^3 - y^3 + (- 27x^2y + 9xy^2)
= (3x - y)^3 - 3xy(3x - y)
So, the factorized form of 27x^3 - 27x^2y + 9xy^2 - y^3 is (3x - y)^3 - 3xy(3x - y).