Consider the following four polynomials, labeled A - D:
A. ( 2x2 - 4x )
B. ( x3 - 1 )
C. ( -3x3 + x2 - 4x )
D. ( x5 + x2 - 3 )
Find an algebraic expression that uses each of the four polynomials once and simplifies to:
x8 + 3
Your expression may involve any combination of adding, subtracting, and multiplying the polynomials. Be sure to explain the thinking you used to find your answer, and show that your expression simplifies correctly.
(x^5+x^2-3)(x^3-1) = x^8-3x^3-x^2+3
Now subtract -3x^3+x^2-4x and you get
x^8-2x^2+4x+3
Now subtract 2x^2-4x and you get
x^8+3
So, the whole thing is just
(x^5+x^2-3)(x^3-1)-(-3x^3+x^2-4x)-(2x^2-4x)
Great answer,
Although I think you meant ADD 2x^2 - 4x to get x^8 + 3.
Nonetheless, thanks.
To find an algebraic expression that uses each of the four polynomials once and simplifies to x^8 + 3, we can combine the polynomials using addition, subtraction, and multiplication.
First, let's look at the exponents in the expression x^8 + 3. The highest exponent is 8, which means we need to include a term with x^8 in our expression. The only polynomial with an x^8 term is D: (x^5 + x^2 - 3).
Next, we can look at the remaining terms needed to simplify to 3. Since we already have a constant term (-3) from polynomial D, we need to find the combination of the remaining polynomials that simplifies to 6.
Let's compare the coefficients of x in the remaining polynomials:
A: -4x
B: 0 (no x terms)
C: -4x
To eliminate the coefficient of x^1 and still have an x^8 term, we can use polynomials A and C. By taking -4x from both A and C, we can cancel out the x term, leaving us with the simplified expression (-8x).
Putting it all together, our expression is:
D + (-8x) = (x^5 + x^2 - 3) - 8x
To simplify this expression further, let's combine like terms:
(x^5 - 8x) + x^2 - 3 = x^5 + x^2 - 8x - 3
Now, let's check if this expression simplifies to x^8 + 3:
x^5 + x^2 - 8x - 3 = (x^5 + x^2 - 8x) - 3 = x^5 + x^2 - 8x - 3
The expression x^5 + x^2 - 8x - 3 is equivalent to x^8 + 3, so our algebraic expression is correct.
Therefore, the expression that uses each of the four polynomials once and simplifies to x^8 + 3 is (x^5 + x^2 - 3) - 8x.
To find an algebraic expression that uses each of the four polynomials once and simplifies to x^8 + 3, we can start by observing the exponents of x in each polynomial.
Polynomial A: 2x^2 - 4x
Polynomial B: x^3 - 1
Polynomial C: -3x^3 + x^2 - 4x
Polynomial D: x^5 + x^2 - 3
We can notice that the highest exponent of x in polynomial A is 2, in polynomial B is 3, in polynomial C is 3, and in polynomial D is 5.
To achieve x^8, we need to multiply x^2 from polynomial A by x^3 from polynomial B, and x^5 from polynomial D by x^3 from polynomial C.
So far, we have the following expression:
(x^2 * x^3) + (x^5 * x^3) + ...
To simplify further, let's multiply the remaining terms together:
(x^2 * x^3) + (x^5 * x^3) + (-3x^3 + x^2 - 4x)
Multiplying the first two terms:
x^(2+3) + x^(5+3) + (-3x^3 + x^2 - 4x)
Simplifying the exponents:
x^5 + x^8 + (-3x^3 + x^2 - 4x)
Now, we need to ensure that the expression simplifies to 3 by adding or subtracting the remaining terms.
Adding the constant terms:
x^5 + x^8 + (-3x^3 + x^2 - 4x) + (-1 - 3)
Simplifying the expression further:
x^8 + x^5 - 3x^3 + x^2 - 4x - 4
We have successfully found an algebraic expression that uses each of the four polynomials once and simplifies to x^8 + 3:
x^8 + x^5 - 3x^3 + x^2 - 4x - 4