Understanding the Relationship Between the Trinomial and the Binomial

Oct 13, 2025 by magicalceylontours.com 126 views

Understanding the Relationship Between the Trinomial and the Binomial

The given trinomial, $$x^2 + 16x + 63$$, represents the area of a rectangle. To find the binomial that could represent the width of the rectangle, we need to understand the relationship between the trinomial and the binomial. The trinomial can be factored into the product of two binomials, which can be represented as:

(x + a)(x + b)

where $a$ and $b$ are the roots of the trinomial.

Factoring the Trinomial

To factor the trinomial, we need to find two numbers whose product is $63$ and whose sum is $16$ . These numbers are $7$ and $9$ , since $7 \times 9 = 63$ and $7 + 9 = 16$ .

Therefore, the trinomial can be factored as:

(x + 7)(x + 9)

Identifying the Binomial Representation of the Width

Since the trinomial represents the area of the rectangle, the binomial representation of the width can be obtained by dividing the trinomial by the binomial representation of the length. In this case, the binomial representation of the length is $$(x + 9)$$.

Therefore, the binomial representation of the width is:

(x + 7)

Conclusion

The binomial that could represent the width of the rectangle is $$(x + 7)$$. This is because the trinomial $$x^2 + 16x + 63$$ can be factored into the product of two binomials, $$(x + 7)(x + 9)$$, where $$(x + 7)$$ represents the width of the rectangle.

Answer

The correct answer is:

A. $$(x + 7)$$

Explanation

The other options, $$(x - 3)$$, $$(x + 9)$$, and $$(x - 9)$$, do not represent the width of the rectangle. The correct binomial representation of the width is $$(x + 7)$$, which is obtained by factoring the trinomial $$x^2 + 16x + 63$$ into the product of two binomials.

Mathematical Representation

The trinomial $$x^2 + 16x + 63$$ can be represented mathematically as:

(x + 7)(x + 9)

where $$(x + 7)$$ represents the width of the rectangle and $$(x + 9)$$ represents the length of the rectangle.

Conclusion

Q: What is the relationship between the trinomial and the binomial?

A: The trinomial can be factored into the product of two binomials, which can be represented as:

(x + a)(x + b)

where $a$ and $b$ are the roots of the trinomial.

Q: How do I factor the trinomial?

A: To factor the trinomial, you need to find two numbers whose product is the constant term (in this case, 63) and whose sum is the coefficient of the middle term (in this case, 16). These numbers are 7 and 9, since 7 × 9 = 63 and 7 + 9 = 16.

Q: What is the binomial representation of the width?

A: The binomial representation of the width can be obtained by dividing the trinomial by the binomial representation of the length. In this case, the binomial representation of the length is (x + 9).

Q: Why is (x + 7) the correct answer?

A: (x + 7) is the correct answer because it represents the width of the rectangle. The trinomial x^2 + 16x + 63 can be factored into the product of two binomials, (x + 7)(x + 9), where (x + 7) represents the width of the rectangle.

Q: What are the other options, and why are they incorrect?

A: The other options, (x - 3), (x + 9), and (x - 9), do not represent the width of the rectangle. They are incorrect because they do not satisfy the condition of being the product of two binomials that can be factored into the trinomial x^2 + 16x + 63.

Q: How can I represent the trinomial mathematically?

A: The trinomial x^2 + 16x + 63 can be represented mathematically as:

(x + 7)(x + 9)

where (x + 7) represents the width of the rectangle and (x + 9) represents the length of the rectangle.

Q: What is the significance of the trinomial representation of a rectangle's area?

A: The trinomial representation of a rectangle's area is significant because it allows us to understand the relationship between the area of a rectangle and its dimensions. It also provides a mathematical framework for solving problems involving rectangles and their areas.

Q: How can I apply this knowledge to real-world problems?

A: You can apply this knowledge to real-world problems by using the trinomial representation of a rectangle's area to solve problems involving rectangles and their areas. For example, you can use this knowledge to calculate the area of a rectangle given its dimensions, or to find the dimensions of a rectangle given its area.

Conclusion

In conclusion, the trinomial representation of a rectangle's area is a powerful tool for understanding the relationship between the area of a rectangle and its dimensions. By factoring the trinomial into the product of two binomials, we can identify the binomial representation of the width and length of the rectangle. This knowledge can be applied to real-world problems involving rectangles and their areas.