Use Pascal's triangle to expand the expression.
(x − y)5
so I tried to work this out:
(x-y)^5 = [(x-y)(x-y)][(x-y)(x-y)](x-y)
(x-y)^2 = (x-y)(x-y) = x^2 - 2xy + y^2
x^2 - 2xy + y^2)(x^2 - 2xy + y^2) = x^4 -4x^3y + 6x^2y^2 - 4xy^3 + y^4
I am still confused can any body help me out. ?
In Pascal's Triangle, the coefficients on the 5th row are 1,5,10,10,5,1
So,
(x-y)^5 = x^5 + 5(x^4)(-y) + 10(x^3)(-y)^2 + 10(x^2)(-y)^3 + 5x(-y)^4 + (-5)^5
= x^5 - 5x^4y + 10x^3y^2 - 10x^2y^3 + 5xy^4 - y^5
To expand the expression (x - y)^5 using Pascal's triangle, you can follow these steps:
1. Start by writing down the coefficients from the fifth row of Pascal's triangle. The fifth row consists of the numbers 1, 4, 6, 4, and 1. These numbers represent the coefficients of the terms in the expansion.
2. The exponent of x starts at 5 and decreases by 1 with each term, while the exponent of y starts at 0 and increases by 1 with each term.
3. Write down the terms of the expansion, keeping in mind the coefficients from the Pascal's triangle and the exponents of x and y. Multiply each term by its respective coefficient and combine like terms.
Let's go through the steps:
Step 1: Write down the coefficients from the fifth row of Pascal's triangle: 1, 4, 6, 4, 1.
Step 2: Start with the largest exponent of x (x^5) and the smallest exponent of y (y^0). The exponents of x decrease by 1 with each term, and the exponents of y increase by 1 with each term.
(x^5)(y^0) = 1*x^5*y^0 = x^5
Step 3: Multiply the first term, x^5, by the first coefficient, 1: 1 * x^5 = x^5.
Now, let's move on to the second term:
(x^4)(y^1) = 4*x^4*y^1 = 4*x^4*y
Step 4: Multiply the second term, 4*x^4*y, by the second coefficient, 4: 4 * (4*x^4*y) = 4*x^4*y^1 = 4*x^4*y.
Next term:
(x^3)(y^2) = 6*x^3*y^2 = 6*x^3*y^2
Multiply the third term, 6*x^3*y^2, by the third coefficient, 6: 6 * (6*x^3*y^2) = 6*x^3*y^2 = 36*x^3*y^2.
Next term:
(x^2)(y^3) = 4*x^2*y^3 = 4*x^2*y^3
Multiply the fourth term, 4*x^2*y^3, by the fourth coefficient, 4: 4 * (4*x^2*y^3) = 4*x^2*y^3 = 16*x^2*y^3.
Final term:
(x^1)(y^4) = 1*x^1*y^4 = x*y^4
Multiply the fifth term, x*y^4, by the fifth coefficient, 1: 1 * (x*y^4) = x*y^4 = x*y^4.
Now, combine all the terms:
x^5 - 4*x^4*y + 6*x^3*y^2 - 4*x^2*y^3 + x*y^4
Therefore, the expanded form of (x - y)^5 is x^5 - 4*x^4*y + 6*x^3*y^2 - 4*x^2*y^3 + x*y^4.