Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Use Pascal's Triangle to expand
(z^(2)+5x^(2))^(4). Express your answer in simplest form.
Answer:

Use Pascal's Triangle to expand $\left(z^{2}+5 x^{2}\right)^{4}$ . Express your answer in simplest form. $\newline$ Answer:

View step-by-step help

View step-by-step help

Pascal's triangle and the Binomial Theorem

Full solution

Q. Use Pascal's Triangle to expand

\left(z^{2}+5 x^{2}\right)^{4}

. Express your answer in simplest form.

\newline

Answer:

Identify Row for Exponent $4$ : Identify the row of Pascal's Triangle that corresponds to the exponent $4$ . The row for the exponent $4$ in Pascal's Triangle is the fifth row (starting with row $0$ for the exponent $0$ ), which has the coefficients $1, 4, 6, 4, 1$ .
Write Binomial Expansion: Write out the terms using the binomial expansion theorem with the coefficients from Pascal's Triangle. $\newline$ The binomial expansion of $(a+b)^4$ is given by: $\newline$ $1\cdot a^4 + 4\cdot a^3\cdot b + 6\cdot a^2\cdot b^2 + 4\cdot a\cdot b^3 + 1\cdot b^4$ $\newline$ Substitute $a$ with $z^2$ and $b$ with $5x^2$ to get: $\newline$ $1\cdot (z^2)^4 + 4\cdot (z^2)^3\cdot (5x^2) + 6\cdot (z^2)^2\cdot (5x^2)^2 + 4\cdot (z^2)\cdot (5x^2)^3 + 1\cdot (5x^2)^4$
Simplify Each Term: Simplify each term. $\newline$ Now we simplify each term: $\newline$ $1*(z^2)^4 = z^8$ $\newline$ $4*(z^2)^3*(5x^2) = 4*z^6*5x^2 = 20z^6x^2$ $\newline$ $6*(z^2)^2*(5x^2)^2 = 6*z^4*(5x^2)^2 = 6*z^4*25x^4 = 150z^4x^4$ $\newline$ $4*(z^2)*(5x^2)^3 = 4*z^2*(5x^2)^3 = 4*z^2*125x^6 = 500z^2x^6$ $\newline$ $1*(5x^2)^4 = (5x^2)^4 = 625x^8$
Combine Final Expanded Form: Combine all the terms to write the final expanded form. $\newline$ The final expanded form is: $\newline$ $z^8 + 20z^6x^2 + 150z^4x^4 + 500z^2x^6 + 625x^8$

More problems from Pascal's triangle and the Binomial Theorem

Question

Find the degree of this polynomial.

\newline

`5b^{10} + 3b^3`

\newline

Posted 2 months ago

Question

\newline

(9a + 5) - (2a + 4)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

-3v^2(v^2 - 9)

\newline

Posted 1 month ago

Question

Find the product. Simplify your answer.

\newline

(v - 3)(4v + 1)

\newline

Posted 2 months ago

Question

Find the square. Simplify your answer.

\newline

(3y + 2)^2

\newline

Posted 2 months ago

Question

Divide. If there is a remainder, include it as a simplified fraction.

\newline

(24t^2 + 36t) \div 6t

\newline

Posted 2 months ago

Question

Is the function

q(x) = x^6 - 9

even, odd, or neither?

\newline

\newline

\text{[[even][odd][neither]]}

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

-3q^2(-3q^2 + q)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

(r + 3)(4r + 2)

\newline

Posted 2 months ago

Question

Find the roots of the factored polynomial.

\newline

(x + 7)(x + 4)

\newline

Write your answer as a list of values separated by commas.

\newline

x =

Posted 2 months ago