Problem 6 - Algebra for College Students

Question

Factor completely each of the polynomials and indicate any that are not factorable using integers. $$a^{2}+6 a-40$$

Step-by-Step Solution

Answer

The polynomial \(a^2 + 6a - 40\) factors to \((a + 10)(a - 4)\).

Step 1: Identify the Quadratic Structure

The given polynomial is in the form of a quadratic expression, which we can write as \(a^2 + 6a - 40\). The general form of a quadratic expression is \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 6\), and \(c = -40\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic expression \(ax^2 + bx + c\) is given by \(b^2 - 4ac\). Substituting the values, we get:\[b^2 - 4ac = 6^2 - 4 \times 1 \times (-40) = 36 + 160 = 196.\]Since the discriminant is a perfect square, the quadratic can be factored over the integers.

Step 3: Find Two Numbers that Multiply to c and Add to b

We need to find two numbers whose product is equal to \(c = -40\) and whose sum is equal to \(b = 6\). These numbers are \(10\) and \(-4\), since:\[10 \times (-4) = -40 \,\text{and}\, 10 + (-4) = 6.\]

Step 4: Factor the Quadratic Expression

Using the numbers from Step 3, we can factor the expression \(a^2 + 6a - 40\) as follows:\[(a + 10)(a - 4).\]This is because when you expand \((a + 10)(a - 4)\), you get \(a^2 + 6a - 40\).

Step 5: Verify the Factorization

Let's verify by expanding \((a + 10)(a - 4)\):\[(a + 10)(a - 4) = a(a-4) + 10(a-4) = a^2 - 4a + 10a - 40 = a^2 + 6a - 40.\]The factored expression is correct.

Key Concepts

quadratic expressions

A quadratic expression is a polynomial of degree two, which means it involves a variable raised to the second power. It typically takes the form \(ax^2 + bx + c\). In a quadratic expression, \(a\), \(b\), and \(c\) are coefficients, where \(a\) must not be zero. Quadratic expressions form a significant part of algebra and have many applications in various fields, including physics and engineering.

• The term \(ax^2\) is the quadratic term, indicating the parabolic shape of the graphed function.
• The \(bx\) term is the linear component.
• \(c\) represents the constant or y-intercept when the expression is graphed.

To manipulate these expressions, we often convert them into their factorized forms, which allow for easier graphing and solving.”

discriminant

The discriminant is a crucial tool in determining the nature of the roots of a quadratic expression. Calculated as \(b^2 - 4ac\) for an expression \(ax^2+bx+c\), its value can inform us about the number and type of solutions available.

• If the discriminant is positive and a perfect square, as in our exercise where \(b^2 - 4ac = 196\), there are two rational roots.
• If it is positive but not a perfect square, the quadratic has two irrational roots.
• A zero discriminant indicates a perfect square, resulting in exactly one real root, which is repeated.
• A negative discriminant signifies imaginary roots, meaning the quadratic does not factor over the real numbers.

Understanding the discriminant helps in anticipating the factorability of the quadratic.”

integer factorization

Integer factorization involves expressing a polynomial using integers. For quadratics, this means rewriting it in the form \((px + q)(rx + s)\) where \(p\), \(q\), \(r\), and \(s\) are integers. This is possible when the quadratic expression can be decomposed into factors whose coefficients are integers.

• Identify the numbers that multiply to the constant term, \(c\), and add to the linear coefficient, \(b\).
• In our example, these numbers are \(10\) and \(-4\), because \(10 \times (-4) = -40\) and \(10 + (-4) = 6\).
• Thus, the polynomial \(a^2 + 6a - 40\) factors as \((a + 10)(a - 4)\).

This factorization allows for solutions to the equation by setting each factor equal to zero, demonstrating the power of integer factorization in solving quadratic expressions.”