The expression $16x^2 + 48x + 36$ is a quadratic polynomial. Quadratic polynomials are algebraic expressions of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our specific case, $a = 16$, $b = 48$, and $c = 36$
Factoring the Polynomial
One way to understand this quadratic polynomial is to factor it. Factoring involves rewriting the polynomial as a product of simpler expressions. Let’s factor $16x^2 + 48x + 36$ step-by-step:
Identify the Greatest Common Factor (GCF): In this case, the GCF of $16x^2$, $48x$, and $36$ is 4. However, we will see that the polynomial can be factored further.
Rewrite the Polynomial: Notice that $16x^2$, $48x$, and $36$ are perfect squares or products of perfect squares. This suggests that the polynomial might be a square of a binomial. Let’s test this hypothesis.
Find the Binomial: We look for a binomial $(ax + b)$ such that $(ax + b)^2 = 16x^2 + 48x + 36$
Solve for $a$ and $b$:
- The first term $16x^2$ suggests that $a = 4$ because $(4x)^2 = 16x^2$
- The constant term $36$ suggests that $b = 6$ because $6^2 = 36$
- Verify the middle term $48x$: $2ab = 2(4x)(6) = 48x$
So, the binomial is $(4x + 6)$, and the quadratic polynomial can be factored as:
$(16x^2 + 48x + 36) = (4x + 6)^2$
Graphical Interpretation
When you graph the quadratic polynomial $16x^2 + 48x + 36$, it forms a parabola. Since the quadratic is a perfect square, the vertex of the parabola touches the x-axis at a single point. The vertex form of a quadratic equation is given by $a(x – h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
For $16x^2 + 48x + 36$, rewriting it in vertex form gives us:
$(4x + 6)^2 = 16(x + frac{3}{2})^2$
Thus, the vertex of the parabola is at $(-frac{3}{2}, 0)$. This means the graph of the polynomial touches the x-axis at $x = -frac{3}{2}$ and opens upwards because the coefficient of $x^2$ (16) is positive.
Conclusion
In summary, the expression $16x^2 + 48x + 36$ is a quadratic polynomial that can be factored into $(4x + 6)^2$. This means it represents a parabola with a vertex at $(-frac{3}{2}, 0)$, which opens upwards. Understanding how to factor and graph such polynomials is fundamental in algebra and helps in solving quadratic equations and analyzing their properties.