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What Is The Product Of (2p + 7)(3p2 + 4p 3)?

What Is The Product Of (2p + 7)(3p2 + 4p 3)?

Understanding how to multiply algebraic expressions like (2p + 7)(3p2 + 4p 3) is fundamental in algebra. This process involves applying the distributive property, often called the FOIL method for binomials, to simplify expressions into a single polynomial. Whether you're a student brushing up on algebra skills or someone seeking to deepen your understanding of polynomial multiplication, this article provides a comprehensive guide to calculating the product of these two binomials, along with explanations, tips, and detailed steps.

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Breaking Down the Expression: (2p + 7)(3p2 + 4p 3)

Before diving into the multiplication process, it’s essential to analyze the components:


  • The first binomial is (2p + 7).

  • The second binomial is (3p2 + 4p 3).


Note: The notation 3p2 typically indicates 3p^2 (3 times p squared), and 4p 3 likely means 4p + 3 (4 times p plus 3). Clarifying this is crucial because the interpretation affects the multiplication process.

Assumption:


  • 3p2 = 3p²

  • 4p 3 = 4p + 3


This interpretation aligns with standard algebraic notation, where:

  • Exponent notation is used with a caret or superscript (e.g., p²).

  • When written without an exponent symbol, "p2" is often read as p squared.


Rewritten Expression for Clarity:
(2p + 7) (3p² + 4p + 3)

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Understanding Polynomial Multiplication

Multiplying two binomials or polynomials involves applying the distributive property systematically. The key steps include:


  1. Distribute each term in the first binomial over each term in the second.

  2. Multiply coefficients and variables separately.

  3. Combine like terms to simplify the expression.


Why is this important?
Mastering these steps ensures accuracy in solving algebraic expressions, which is vital for higher-level math topics like quadratic equations, calculus, and beyond.

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Step-by-Step Guide to Multiply (2p + 7)(3p² + 4p + 3)

Step 1: Distribute 2p over the second polynomial

Multiply 2p by each term in (3p² + 4p + 3):


  • 2p 3p² = 2 3 p p² = 6p³

  • 2p 4p = 2 4 p p = 8p²

  • 2p 3 = 2 3 p = 6p


Step 2: Distribute 7 over the second polynomial

Multiply 7 by each term in (3p² + 4p + 3):


  • 7 3p² = 21p²

  • 7 4p = 28p

  • 7 3 = 21


Step 3: Write the expanded terms

Combine all results:


  • 6p³

  • 8p²

  • 6p

  • 21p²

  • 28p

  • 21


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Combining Like Terms

Now, group similar terms to simplify the expression:


  • p³ term: 6p³ (only one)

  • p² terms: 8p² + 21p² = (8 + 21) p² = 29p²

  • p terms: 6p + 28p = (6 + 28) p = 34p

  • Constant term: 21


Final simplified expression:

6p³ + 29p² + 34p + 21

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Complete Polynomial Product

Answer:
The product of (2p + 7) and (3p² + 4p + 3) is:

6p³ + 29p² + 34p + 21

This cubic polynomial represents the expanded form of the multiplication.

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Additional Tips for Polynomial Multiplication

Use of Distributive Property


  • Always distribute each term in the first binomial across all terms in the second.

  • Keep track of signs (+ or -) to avoid errors.


Combining Like Terms

  • Group similar powers of p to simplify your expression.

  • Double-check coefficients for accuracy.


Organizing Your Work

  • Write each step clearly.

  • Use a table or grid if needed to keep track of multiplications.


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Practice Problems for Mastery

  1. Multiply (p + 4)(2p² + p + 5)
  2. Find the product of (3p - 2)(p² + 4p + 1)
  3. Expand (5p + 3)(p² - 2p + 7)
Tip: Always follow the same step-by-step method: distribute each term, multiply coefficients and variables, then combine like terms.

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Common Mistakes to Avoid

  • Misinterpreting the notation: Confirm whether variables are squared or multiplied.
  • Forgetting to distribute all terms: Ensure every term in the first binomial multiplies every term in the second.
  • Incorrectly combining like terms: Only combine terms with the same variable and exponent.
  • Sign errors: Pay attention to positive and negative signs during distribution.
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Real-World Applications of Polynomial Multiplication

Polynomial multiplication is crucial in various fields:


  • Engineering: Calculating areas, volumes, and stress analysis.

  • Physics: Expressing equations of motion or wave functions.

  • Computer Science: Algorithms involving polynomial operations.

  • Economics: Modeling cost and revenue functions.


Understanding how to multiply polynomials efficiently allows professionals and students to model and solve complex real-world problems accurately.

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Summary

In conclusion, multiplying (2p + 7)(3p² + 4p + 3) involves applying the distributive property to each term, multiplying coefficients and variables carefully, and then combining like terms to arrive at the simplified polynomial expression. The final result is 6p³ + 29p² + 34p + 21. Mastery of this process is essential in algebra and forms the foundation for more advanced mathematical concepts.

By practicing with similar problems and following systematic steps, you'll become proficient in polynomial multiplication, enabling you to tackle a wide range of algebraic challenges with confidence.

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Remember: Always analyze the expression carefully, interpret notation properly, and work systematically to ensure accuracy in your algebraic calculations.

Frequently Asked Questions

What is the product of (2p + 7) and (3p^2 + 4p^3)?
The product is obtained by distributing each term: (2p)(3p^2 + 4p^3) + 7(3p^2 + 4p^3), which simplifies to 6p^3 + 8p^4 + 21p^2 + 28p^3.
How do you expand the expression (2p + 7)(3p^2 + 4p^3)?
Expand by distributing each term: multiply 2p by each term in the second polynomial and then 7 by each term, then combine like terms.
What is the simplified form of (2p + 7)(3p^2 + 4p^3)?
The simplified form is 8p^4 + 28p^3 + 21p^2.
Can you factor the expression (2p + 7)(3p^2 + 4p^3)?
Yes, factoring is not necessary here since it's a product of two polynomials, but after expansion, the polynomial can sometimes be factored further depending on context.
What degree polynomial is obtained after multiplying (2p + 7) and (3p^2 + 4p^3)?
The resulting polynomial is of degree 4, with the highest power term being 8p^4.
What are the coefficients in the expanded form of (2p + 7)(3p^2 + 4p^3)?
The coefficients are 8 for p^4, 28 for p^3, and 21 for p^2.
Is the multiplication of (2p + 7)(3p^2 + 4p^3) commutative?
No, the multiplication of polynomials is not commutative in the sense of changing the order of factors, but the product is the same regardless of the order of multiplication.
What is the importance of understanding polynomial multiplication like (2p + 7)(3p^2 + 4p^3)?
Understanding polynomial multiplication is fundamental in algebra, allowing for the expansion, simplification, and factoring of complex expressions, which are essential skills in higher mathematics and problem-solving.

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