Use suitable identity to get product:
1) (x3 - 3/8) (x3 + 1/4)
2) (p2 + 16) (p2 - 1/4)
3) (y - 7) (y + 12)
Thanks in anticipation.
I don't understand the instruction, why don't we just go ahead and do the problems
e.g.
#2
(p^2 + 16)(p^2 - 1/4)
= p^4 - (1/4)p^2 + 16p^2 - 4
= p^4 -(63/4)p^2 - 4
It is16
13'6"
To simplify these expressions, we can use the identity of the difference of squares.
The identity of the difference of squares states that for any two values a and b, the product (a + b)(a - b) can be simplified to (a^2 - b^2).
Let's apply this identity to the given expressions:
1) (x^3 - 3/8)(x^3 + 1/4)
We have the difference of cubes here, where a = x^3 and b = 3/8. To simplify, we'll use the identity (a^3 - b^3) = (a - b)(a^2 + ab + b^2)
Therefore, applying the identity:
(x^3 - 3/8)(x^3 + 1/4)
= [(x^3) - (3/8)][(x^3) + (1/4)]
= [x^3)^2 - (3/8)((x^3) + (1/4))(x^3)] + [(3/8)(1/4)]
= [x^6 - (3/8)(x^3)] + [(3/32)]
= x^6 - (3/8)x^3 + (3/32)
2) (p^2 + 16)(p^2 - 1/4)
Here, we have the difference of squares. The identity (a^2 - b^2) = (a + b)(a - b) can be applied here.
Therefore, applying the identity:
(p^2 + 16)(p^2 - 1/4)
= [(p^2)^2 - (1/4)^2]
= [p^4 - (1/16)]
3) (y - 7)(y + 12)
This expression doesn't fit any specific identity but can still be simplified through the distributive property.
Using the distributive property, we multiply each term of the first expression (y - 7) by each term of the second expression (y + 12).
= [y * y] + [y * 12] + [-7 * y] + [-7 * 12]
= y^2 + 12y - 7y - 84
= y^2 + 5y - 84
So, the simplified forms of the given expressions are:
1) x^6 - (3/8)x^3 + (3/32)
2) p^4 - (1/16)
3) y^2 + 5y - 84