Type The Correct Answer In Each Box. Use Numerals Instead Of Words.Simplify The Following Polynomial Expression:$\left(5x^2 + 13x - 4\right) - \left(17x^2 + 7x - 19\right) + (5x - 7)(3x + 1$\]$\square X^2 - \square X + \square$

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Understanding the Problem

To simplify the given polynomial expression, we need to apply the rules of algebra, specifically the distributive property and combining like terms. The expression is given as:

(5x2+13x−4)−(17x2+7x−19)+(5x−7)(3x+1)\left(5x^2 + 13x - 4\right) - \left(17x^2 + 7x - 19\right) + (5x - 7)(3x + 1)

Our goal is to simplify this expression and write it in the form ax2+bx+cax^2 + bx + c.

Step 1: Distribute the Terms

First, we need to distribute the terms in the last part of the expression, (5x−7)(3x+1)(5x - 7)(3x + 1). This can be done using the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b + c) = ab + ac.

(5x - 7)(3x + 1) = 5x(3x + 1) - 7(3x + 1)

Step 2: Apply the Distributive Property

Now, we can apply the distributive property to each term:

5x(3x + 1) = 15x^2 + 5x
- 7(3x + 1) = -21x - 7

Step 3: Combine Like Terms

Next, we need to combine like terms in the expression. We can start by combining the x2x^2 terms, the xx terms, and the constant terms separately.

(5x^2 + 13x - 4) - (17x^2 + 7x - 19) = -12x^2 + 6x + 15
(15x^2 + 5x) - 21x - 7 = 15x^2 - 16x - 7

Step 4: Simplify the Expression

Now, we can simplify the expression by combining the like terms:

(-12x^2 + 6x + 15) + (15x^2 - 16x - 7) = 3x^2 - 10x + 8

Conclusion

The simplified polynomial expression is 3x2−10x+83x^2 - 10x + 8. This is the final answer.

Final Answer

3x2−10x+8\boxed{3x^2 - 10x + 8}

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Understanding the Problem

To simplify the given polynomial expression, we need to apply the rules of algebra, specifically the distributive property and combining like terms. The expression is given as:

(5x2+13x−4)−(17x2+7x−19)+(5x−7)(3x+1)\left(5x^2 + 13x - 4\right) - \left(17x^2 + 7x - 19\right) + (5x - 7)(3x + 1)

Our goal is to simplify this expression and write it in the form ax2+bx+cax^2 + bx + c.

Q&A

Q: What is the first step in simplifying the given polynomial expression?

A: The first step is to distribute the terms in the last part of the expression, (5x−7)(3x+1)(5x - 7)(3x + 1).

Q: How do we apply the distributive property to each term?

A: We apply the distributive property by multiplying each term in the first expression by each term in the second expression.

Q: What are the like terms in the expression?

A: The like terms are the x2x^2 terms, the xx terms, and the constant terms.

Q: How do we combine the like terms?

A: We combine the like terms by adding or subtracting the coefficients of the same variables.

Q: What is the final simplified expression?

A: The final simplified expression is 3x2−10x+83x^2 - 10x + 8.

Common Mistakes

Mistake 1: Not Distributing the Terms

  • Not distributing the terms in the last part of the expression can lead to incorrect simplification.
  • Make sure to apply the distributive property to each term.

Mistake 2: Not Combining Like Terms

  • Not combining like terms can lead to incorrect simplification.
  • Make sure to combine the like terms by adding or subtracting the coefficients of the same variables.

Mistake 3: Not Simplifying the Expression

  • Not simplifying the expression can lead to incorrect simplification.
  • Make sure to simplify the expression by combining the like terms.

Tips and Tricks

Tip 1: Use the Distributive Property

  • The distributive property is a powerful tool for simplifying expressions.
  • Make sure to apply the distributive property to each term.

Tip 2: Combine Like Terms

  • Combining like terms is an essential step in simplifying expressions.
  • Make sure to combine the like terms by adding or subtracting the coefficients of the same variables.

Tip 3: Simplify the Expression

  • Simplifying the expression is the final step in simplifying the given polynomial expression.
  • Make sure to simplify the expression by combining the like terms.

Conclusion

Simplifying the given polynomial expression requires careful application of the distributive property and combining like terms. By following the steps outlined in this article, you can simplify the expression and write it in the form ax2+bx+cax^2 + bx + c. Remember to use the distributive property, combine like terms, and simplify the expression to get the final answer.

Final Answer

3x2−10x+8\boxed{3x^2 - 10x + 8}