Problem 46 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$t^{2}-2 t-143$$

Step-by-Step Solution

Answer

The polynomial factors to \((t + 11)(t - 13)\).

Step 1: Identify Potential Factorization Form

The polynomial given is a quadratic trinomial in the form \( t^2 - 2t - 143 \). Quadratics can often be factored into the product of two binomials: \( (t + a)(t + b) \), where \( a \) and \( b \) are numbers that satisfy specific conditions.

Step 2: Apply the Product and Sum Method

We need to find two numbers that multiply to \(-143\) (the constant term) and add to \(-2\) (the coefficient of the linear term). The product and sum of these numbers should satisfy the following equations: \[ ab = -143 \] \[ a + b = -2 \]

Step 3: Determine Integer Values for a and b

Consider the factors of \(-143\). The pairs of integers that multiply to \(-143\) include \((-11, 13)\) and \((11, -13)\). Check which pair adds up to \(-2\). Check: - \(-11 + 13 = 2\) - \(11 - 13 = -2\)The correct pair is \(11\) and \(-13\).

Step 4: Write the Polynomial as a Product of Binomials

Now that we have found the correct values for \(a\) and \(b\), substitute them back into the binomial form:\[ (t + 11)(t - 13) \]

Step 5: Verify the Factorization

To confirm the factorization is correct, expand \((t + 11)(t - 13)\):- Multiply: \((t + 11)(t - 13) = t^2 - 13t + 11t - 143\)- Simplify: \(t^2 - 2t - 143\)The expanded form matches the original polynomial, confirming the factorization is correct.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a type of polynomial that includes three terms, with the highest degree term being squared. In this case, the polynomial is given as \( t^2 - 2t - 143 \). The general form of a quadratic trinomial is \( ax^2 + bx + c \), where \( a \) is the coefficient of the squared term, \( b \) is the coefficient of the linear term, and \( c \) is the constant. Here, \( a = 1 \), \( b = -2 \), and \( c = -143 \).

Quadratic trinomials can often be rewritten as the product of two binomials if suitable integers are found that meet specific conditions related to \( b \) and \( c \). Recognizing this structure is crucial because it allows us to factor the trinomial effectively by splitting it into simpler components. This makes solving equations and simplifying expressions much easier.

Binomial Product

A binomial product results from multiplying two binomials together. In simpler terms, a binomial is a polynomial with two terms, like \( (t + a) \) and \( (t + b) \).

When you're asked to factor a quadratic trinomial such as \( t^2 - 2t - 143 \), you're essentially looking to express it as a binomial product. This process involves finding two binomials whose product is the original trinomial.

Our aim is to find numbers that when added give \( -2 \) and when multiplied give \( -143 \). Once these numbers \( a \) and \( b \) are identified, they are used to construct the binomials. For \( t^2 - 2t - 143 \), these numbers are \( 11 \) and \(-13\), forming the binomial product \((t + 11)(t - 13)\).

Integer Factorization

Integer factorization involves breaking down a number into its integer components. When dealing with polynomials, it means expressing numbers as products of their integer factors.

For the polynomial \( t^2 - 2t - 143 \), we need to factor \(-143\) into two integers that add up to \(-2\). Key integer pairs for \(-143\) include \((-11, 13)\) and \((11, -13)\). By checking these pairs:

• \(-11 + 13 = 2\)
• \(11 - 13 = -2\)

This means the correct pair is \(11\) and \(-13\), making them the necessary integers for the binomial product. This step highlights the importance of integer factorization in polynomial simplification.

Product and Sum Method

The product and sum method is an effective technique for factoring quadratic trinomials. It requires finding two numbers that multiply to the last term of the trinomial (the constant) and add to the middle term coefficient (linear coefficient).

For the problem \( t^2 - 2t - 143 \):

• We need numbers that multiply to \(-143\) (the constant term).
• We need these numbers to add up to \(-2\) (the coefficient of the linear term).

This requires systematically testing factor pairs of \(-143\). The successful pair, in this case, is \(11\) and \(-13\), leading to effective factorization into the binomials \((t + 11)(t - 13)\). Understanding the product and sum method provides a straightforward path to factorization, making polynomial equations simpler to solve.