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When we talk about the C-intercept of a polynomial function like \(C(t) = 4t^4 + 12t^3 - 40t^2\), we refer to the point where the graph of this function intersects the C-axis (similar to the y-axis in the Cartesian plane). To find this intercept, we set the variable \(t\) to zero. This is because the C-intercept represents the value of the function when \(t\) is zero.
Everything narrows down to a simple substitution in our function: substitute \(t = 0\). For this function, it simplifies to \(C(0)= 4(0)^4 + 12(0)^3 - 40(0)^2 = 0\). This calculation demonstrates that the C-intercept is at the origin, or the point \((0, 0)\). So, a C-intercept is essentially the constant term of the polynomial when written in standard form, unless all terms include the variable as in this case.

To determine the t-intercepts, you need to find the values of \(t\) where the function \(C(t)\) equals zero. This means solving the equation \(4t^4 + 12t^3 - 40t^2 = 0\) for \(t\). These are the points where the function crosses the t-axis (or x-axis).
The given polynomial can be factored to make this process easier. First, identify any common factors. In this exercise, the greatest common factor is \(4t^2\), leading to the factoring of the function as \(4t^2(t^2 + 3t - 10) = 0\).

• The term \(4t^2\) gives us a solution \(t = 0\) (already found as a C-intercept too).
• The quadratic \(t^2 + 3t - 10 = 0\) leads to finding the remaining t-intercepts utilizing the quadratic formula.

The quadratic formula is a fundamental technique used to find the roots of a quadratic equation, commonly expressed as \(ax^2 + bx + c = 0\). In our exercise, once we factor out \(4t^2\), we are left with solving \(t^2 + 3t - 10 = 0\). To do this, the quadratic formula comes to the rescue.
The formula itself is a precise calculation expressed as:\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here \(a = 1\), \(b = 3\), and \(c = -10\). Start with the discriminant \(b^2 - 4ac\), where \(b^2 = 9\) and \(-4ac = 40\), resulting in \(49\). The discriminant is positive, indicating two real solutions.
Solving this provides the roots \(t = \frac{-3 \pm 7}{2}\), which simplifies to \(t = 2\) and \(t = -5\). These solutions correspond to the other t-intercepts \((2, 0)\) and \((-5, 0)\) on the t-axis.

Factoring polynomials involves breaking down a polynomial into the product of simpler polynomials. It's a crucial skill that simplifies solving equations, finding roots, and understanding functions. In our example, the function \(C(t) = 4t^4 + 12t^3 - 40t^2\) can be made more manageable through factoring.
Recognize the greatest common factor (GCF) across terms, which is \(4t^2\). Factoring this out gives us:
\[(4t^2)(t^2 + 3t - 10) = 0\]This step decomposes the original polynomial into a product of \(4t^2\) and a quadratic \(t^2 + 3t - 10\).

• Factoring out \(4t^2\) immediately gives the solution \(t = 0\).
• The remaining quadratic can be solved further using the quadratic formula.

Factoring not only helps in breaking down the expression but also in quickly identifying some specific roots, especially when the polynomial is set equal to zero.