Multiply Each Term in the First Expression by Each Term in the Second Expression

Oct 13, 2025 by windailys.com 23 views

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Introduction

In algebra, expanding an expression to a polynomial in standard form is a crucial skill that helps us simplify complex expressions and solve equations. In this article, we will learn how to expand the given expression $\left(x^2+2x-2\right)\left(3x^2-x-9\right)$ to a polynomial in standard form.

What is a Polynomial in Standard Form?

A polynomial in standard form is an expression of the form $ax^n + bx^{n-1} + \cdots + cx + d$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $n$ is a non-negative integer. The degree of the polynomial is the highest power of the variable, which in this case is $n$ .

Step 1: Multiply Each Term in the First Expression by Each Term in the Second Expression

To expand the given expression, we need to multiply each term in the first expression by each term in the second expression. This will result in a sum of products, which we will then simplify.

Multiply Each Term in the First Expression by Each Term in the Second Expression

$\left(x^2+2x-2\right)\left(3x^2-x-9\right)$

$= \left(x^2\right)\left(3x^2\right) + \left(x^2\right)\left(-x\right) + \left(x^2\right)\left(-9\right) + \left(2x\right)\left(3x^2\right) + \left(2x\right)\left(-x\right) + \left(2x\right)\left(-9\right) + \left(-2\right)\left(3x^2\right) + \left(-2\right)\left(-x\right) + \left(-2\right)\left(-9\right)$

Step 2: Simplify Each Product

Now that we have multiplied each term in the first expression by each term in the second expression, we need to simplify each product.

Simplify Each Product

$\left(x^2\right)\left(3x^2\right) = 3x^4$

$\left(x^2\right)\left(-x\right) = -x^3$

$\left(x^2\right)\left(-9\right) = -9x^2$

$\left(2x\right)\left(3x^2\right) = 6x^3$

$\left(2x\right)\left(-x\right) = -2x^2$

$\left(2x\right)\left(-9\right) = -18x$

$\left(-2\right)\left(3x^2\right) = -6x^2$

$\left(-2\right)\left(-x\right) = 2x$

$\left(-2\right)\left(-9\right) = 18$

Step 3: Combine Like Terms

Now that we have simplified each product, we need to combine like terms.

Combine Like Terms

$3x^4 - x^3 - 9x^2 + 6x^3 - 2x^2 - 18x - 6x^2 + 2x + 18$

$= 3x^4 + 5x^3 - 17x^2 - 16x + 18$

Conclusion

In this article, we learned how to expand the given expression $\left(x^2+2x-2\right)\left(3x^2-x-9\right)$ to a polynomial in standard form. We multiplied each term in the first expression by each term in the second expression, simplified each product, and combined like terms to get the final result.

Final Answer

The final answer is $\boxed{3x^4 + 5x^3 - 17x^2 - 16x + 18}$ .

Example Use Case

Expanding expressions to polynomials in standard form is a crucial skill in algebra that helps us simplify complex expressions and solve equations. For example, if we want to find the value of the expression $\left(x^2+2x-2\right)\left(3x^2-x-9\right)$ when $x = 2$ , we can substitute $x = 2$ into the expanded expression $3x^4 + 5x^3 - 17x^2 - 16x + 18$ to get the final result.

Tips and Tricks

When expanding expressions to polynomials in standard form, make sure to multiply each term in the first expression by each term in the second expression.
Simplify each product by combining like terms.
Combine like terms to get the final result.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Linear Algebra" by Jim Hefferon

Note: The references provided are for example purposes only and are not actual references used in this article.

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Introduction

In our previous article, we learned how to expand the expression $\left(x^2+2x-2\right)\left(3x^2-x-9\right)$ to a polynomial in standard form. In this article, we will answer some frequently asked questions about expanding expressions to polynomials in standard form.

Q1: What is the purpose of expanding expressions to polynomials in standard form?

A1: The purpose of expanding expressions to polynomials in standard form is to simplify complex expressions and solve equations. By expanding expressions to polynomials in standard form, we can easily identify the degree of the polynomial, the leading coefficient, and the constant term.

Q2: How do I know when to expand an expression to a polynomial in standard form?

A2: You should expand an expression to a polynomial in standard form when you need to simplify a complex expression or solve an equation. For example, if you have an expression like $\left(x^2+2x-2\right)\left(3x^2-x-9\right)$ , you should expand it to a polynomial in standard form to simplify it.

Q3: What are some common mistakes to avoid when expanding expressions to polynomials in standard form?

A3: Some common mistakes to avoid when expanding expressions to polynomials in standard form include:

Not multiplying each term in the first expression by each term in the second expression.
Not simplifying each product by combining like terms.
Not combining like terms to get the final result.

Q4: How do I simplify each product when expanding an expression to a polynomial in standard form?

A4: To simplify each product when expanding an expression to a polynomial in standard form, you should combine like terms. For example, if you have a product like $\left(x^2\right)\left(3x^2\right)$ , you should simplify it to $3x^4$ .

Q5: How do I combine like terms when expanding an expression to a polynomial in standard form?

A5: To combine like terms when expanding an expression to a polynomial in standard form, you should add or subtract the coefficients of the like terms. For example, if you have two terms like $3x^4$ and $5x^4$ , you should combine them to get $8x^4$ .

Q6: What are some real-world applications of expanding expressions to polynomials in standard form?

A6: Some real-world applications of expanding expressions to polynomials in standard form include:

Physics: Expanding expressions to polynomials in standard form is used to describe the motion of objects in terms of position, velocity, and acceleration.
Engineering: Expanding expressions to polynomials in standard form is used to design and analyze electrical circuits, mechanical systems, and other complex systems.
Computer Science: Expanding expressions to polynomials in standard form is used in algorithms and data structures to solve problems efficiently.

Q7: How do I know if an expression can be expanded to a polynomial in standard form?

A7: An expression can be expanded to a polynomial in standard form if it is a product of two or more expressions, and each expression is a polynomial. For example, the expression $\left(x^2+2x-2\right)\left(3x^2-x-9\right)$ can be expanded to a polynomial in standard form.

Q8: What are some common expressions that can be expanded to polynomials in standard form?

A8: Some common expressions that can be expanded to polynomials in standard form include:

$\left(x^2+2x-2\right)\left(3x^2-x-9\right)$
$\left(x^3-2x^2+3x-1\right)\left(x+2\right)$
$\left(x^4+2x^3-3x^2+x+1\right)\left(x-1\right)$